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ELEMENTARY  TREATISE 

0N  iU'V\  x\ ;  r  r.f  i. 


IN  TWO  PARTS. 


THE  FIRST, 

CONTAINING  A  CLEAR  AND  COMPENDIOUS 
VIEW  OF  THE  THEORY \ 

THE  SECOND, 

'  •  \  t  •* 

A  NUMBER  OF  PRACTICAL  PROBLEMS. 

TO  WHICH  ARE  ADDED, 

Solar,  Lunar,  and,  some  other 

ASTRONOMICAL  TABLES. 

BY  JOHN  GUMMERE, 

FELLOW  OF  THE  AMERICAN"  PHILOSOPHICAL  SOCIETY,  AND  CORRESPONDING 
MEMBER  OF  THE  ACADEMY  OF  NATURAL  SCIENCES,  PHILADELPHIA, 

PHILADELPHIA: 

PUBLISHED  BY  KIMBER  &  SIIARPLESS,  NO.  93,  MARKET  STREET. 


.T.  PRISSY  AND  G.  GOODMAN,  PRINTERS. 


Eastern  District  of  Pennsylvania,  to  wit: 

BE  IT  REMEMBERED,  that  on  the  second  day  of  January, 
ln  t^ie  forty- sixt4  year  of  the  Independence  of  the  United 
States  of  America,  A.  D.  1822,  Kimber  &  Sharpless,  of  the  said 
district,  have  deposited  in  this  office,  the  title  of  a  book,  the 
right  whereof  they  claim  as  proprietors,  in  the  words  following,  to  wit: 

“An  Elementary  Treatise  on  Astronomy.  In  two  parts.  The  first,  con¬ 
taining  a  clear  and  compendious  view  of  the  Theory.  The  second, 
a  number  of  Practical  Problems.  To  which  are  added.  Solar,  Lunar, 
and  some  other  Astronomical  Tables.  By  John  Gummere,  Fellow  of 
the  American  Philosophical  Society,  and  Corresponding  Member  of 
the  Academy  of  Natural  Sciences,  Philadelphia.” 

In  conformity  to  the  Act  of  Congress  of  the  United  States,  intituled  “An 
act  for  the  encouragement  of  learning,  by  securing  the  copies  of  maps, 
charts,  and  books,  to  the  authors  and  proprietors  of  such  copies,  during 
the  times  therein  mentioned.”  And  also  to  the  act,  entitled  “  An  act,  sup¬ 
plementary  to  an  act,  entitled,  ‘An  act  for  the  encouragement  of  learning, 
by  securing  the  copies  of  maps,  charts,  and  books,  to  the  authors  and  pro¬ 
prietors  of  such  copies,  during  the  times  therein  mentioned,’  and  extending 
the  benefits  thereof  to  the  arts  of  designing,  engraving,  and  etching  histori¬ 
cal  and  other  prints.” 

D.  CALDWELL, 

Clerk  of  the  Eastern  District  of  Penmylvama. 


PREFACE. 


The  object  in  writing  the  present  Treatise,  has  been 
to  give,  in  a  moderate  compass,  a  methodical  and 
scientific  exhibition  of  the  elementary  principles  of 
Astronomy,  and  to  furnish  the  student  with  Rules  and 
Tables  for  making  some  of  the  more  useful  and  more 
interesting  calculations.  The  work  is  divided  into  two 
Parts;  the  First  containing  the  Theory,  and  the  Se¬ 
cond,  Practical  Problems. 

Particular  attention  has  been  given  to  the  arrange¬ 
ment  of  the  First  Part.  The  different  subjects  are  in¬ 
troduced  in  such  order,  as  to  make  it  unnecessary  for 
the  student  to  anticipate  propositions  in  advance  of 
those  which  he  is  studying.  The  Definitions  are  given 
as  they  are  wanted  in  the  course  of  the  work,  and  after 
previous  investigations  have  served  to  render  them 
easily  understood. 

Astronomy,  when  taken  in  its  whole  extent,  and 
with  all  its  different  methods,  necessarily  forms  a 
large  treatise.  It  is  not  therefore  practicable  to  give 
those  various  methods,  in  a  work  of  the  size  to  which 
it  has  been  thought  proper  to  limit  this.  Neither 
are  they  important,  except  to  those  who  devote  very 
decided  attention  to  this  interesting  science;  and  they 
must  have  recourse  to  more  extended  works.  Most 
students  are  satisfied  with  obtaining  a  correct  gene¬ 
ral  knowledge  of  the  subject,  and  of  the  means  by 
which  the  principal  facts  have  been  discovered  or  can 
be  established,  without  entering  into  all  the  investiga- 


IV 


PREFACE. 


tions  necessary  to  render  those  means  the  most  effica¬ 
cious  in  giving  precision  to  the  results.  In  conformity 
with  these  views,  I  have  seldom  given  more  than  one 
method  of  determining  any  particular  fact,  and  have 
avoided  entering  into  minute  details  that  did  not  ap¬ 
pear  necessary  to  a  propercomprehension  of  the  subject. 
In  the  demonstrations,  the  student  is  supposed  to  he 
acquainted  with  Algebra,  Geometry,  Plane  and  Spheri¬ 
cal  Trigonometry,  and  Conic  Sections,  or  at  least  the 
properties  of  the  Ellipse.  As  many  persons  study 
Astronomy  who  have  no  knowledge  of  the  Differential 
Calculus,  it  has  not  been  used,  though  in  a  few  cases 
it  might  have  been  introduced  with  advantage. 

The  Problems  in  the  Second  Part  are  principally 
for  making  calculations  relative  to  the  Sun,  Moon  and 
Fixed  Stars.  The  Tables  of  the  Sun  and  Moon, 
which  are  used  in  these,  have  been  abridged  from  the 
Tables  of  Delambre  and  Burckhardt,  and  reduced  to 
the  meridian  of  Greenwich.  Although  the  quantities 
are  only  given  to  whole  seconds,  and  several  small 
equations  have  been  omitted,  the  places  of  the  suu  and 
moon,  obtained  from  these  tables  will  be  very  nearly 
correct,  for  any  time  within  the  period  to  which  the 
tables  of  Epochs  extend.*  Rules  are  also  given  for 
obtaining  the  places  and  motions  of  the  sun  and  moon 
for  a  given  time  from  the  Nautical  Almanac.  Each  of 
the  problems  is  illustrated  by  one  wrought  example; 


*  The  small  table  for  reducing  the  moon’s  parallax  is  adapted  to  the  mean 
value  of  the  parallax,  and  has  a  degree  of  accuracy  corresponding  to  the 
other  tables.  As  the  same  table  is  used  for  reducing  the  parallax  when  ob¬ 
tained  more  accurately  from  the  Nautical  Almanac,  it  would  have  been  bet¬ 
ter  to  have  adapted  it  to  different  values  of  the  parallax;  but  this  circum¬ 
stance  was  not  thought  of,  in  time  to  alter  it.  And  as  the  change  in  the 
reduction,  for  a  given  latitude,  is  small,  it  is  not  perhaps  important. 


PREFACE. 


V 


and  to  most  of  them  are  added  one  or  two  unwrought 
questions,  with  the  answers  annexed,  to  serve  as  exer¬ 
cises  for  the  student. 

While  writing  the  present  treatise,  I  have  had  re¬ 
course  to  several  of  the  best  modern  publications  on 
the  subject;  among  which  may  be  particularly  mention¬ 
ed  those  of  Vince ,  Wuodhonse,  Playfair ,  Delambre , 
Biot ,  and  Laplace.  From  these  I  have  adopted  the 
methods  which  best  suited  my  purpose,  making,  when 
it  appeared  necessary,  such  modifications  in  them  as  the 
plan  of  the  work  required.  In  the  Projection  of  Eclipses 
and  Occultations,  a  method  is  given  which  is  believed 
to  be  new,  and  which  renders  the  operation  more  sim¬ 
ple,  without  materially  affecting  the  accuracy  of  the 
results.  An  easy  method,  derived  from  the  former,  is 
also  given  for  tracing  the  central  path  of  an  eclipse  of 
the  sun. 

In  a  work  of  this  description,  particularly  when 
printed  from  manuscript,  errors  must  be  expected 
to  occur;  some  proceeding  from  inadvertencies  on  the 
part  of  the  author,  and  others  occurring  in  the  press. 
Such  as  have  been  discovered,  which,  it  is  believed, 
include  all  that  are  important,  are  enumerated  at  the 
end  of  the  volume. 

JOHN  GUMMERE. 

Burlington ,  JV*.  J. 

1 2mo.  22d ,  1821. 


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CONTENTS 


PART  I. 


PAGE. 

Chap.  I.  General  Phenomena  of  the  Heavens,  1 

Chap.  II.  Definitions  of  Terms. — Astronomical  Instruments,  4 

Chap.  III.  Meridian  Line. — Sidereal  Day. — Diurnal  Mo¬ 
tion. — Refraction,  -  -  -  -  7 

Chap.  IV.  Latitude  of  a  Place. — Figure  and  Extent  of  the 

Earth. — Longitude,  -  -  -  -  16 

Chap.  V.  On  Parallax,  22 

Chap.  VI.  Apparent  Path  of  the  Sun. — Fixed  Stars,  -  29 

Chap.  VII.  Sun’s  Apparent  Orbit. — Kepler’s  Laws. — Kep¬ 
ler’s  Problem,  46 

Chap.  VIII.  Equation  of  Time. — Right  Ascension  of  Mid- 

Heaven,  -  -  -  -  71 

Chap.  IX.  Circumstances  of  the  Diurnal  Motion. — Sun’s 

Spots,  and  Rotation  on  its  Axis. — Zodiacal  Light,  77 
Chap.  X.  Of  the  Moon,  -  -  -  -  91 

Chap.  XI.  Eclipses  of  the  Sun  and  Moon. — Occultations,  115 
Chap.  XII.  Of  the  Planets,  -  -  -  155 

Chap.  XIII.  On  Comets,  -  -  -  190 

Chap.  XIV.  Aberration  of  Light. — Nutation  of  the  Earth’s 

Axis. — Annual  Parallax  of  the  Fixed  Stars,  -  193 

Chap.  XV.  Nautical  Astronomy,  -  -  19S 

Chap.  XVI.  Of  the  Calendar,  -  207 

Chap.  XVII.  Universal  Gravitation,  and  some  of  its  Effects,  213 

PART  II. 

Catalogue  of  the  Tables,  -  245 

Observations  and  Rules  relative  to  Quantities  with  different 

signs,  -  -  -  -  250 

PitOB.  I.  To  work  a  proportion  by  logistical  logarithms,  254 
Prob.  II.  From  a  table  in  which  quantities  are  given,  for 
each  Sign  and  Degree  of  the  Circle,  to  find  the  quan¬ 
tity  corresponding  to  Signs,  Degrees,  Minutes  and 
Seconds,  -----  255 

Prob.  III.  To  convert  Degrees,  Minutes  and  Seconds  of  the 

Equator  into  Time,  -  257 


via 


CONTENTS. 


PA  OF. 

Prob.  TV.  To  convert  Time  into  Degrees,  Miuutes  and 

Seconds,  -  -  257 

Prob.  V.  The  longitudes  of  two  Places,  and  the  Time  at  one 
of  them  being  given,  to  find  the  corresponding  Time 
at  the  other,  -----  258 

Prob.  VI  To  convert  apparent  time  into  mean,  and  the  con¬ 
trary,  -  -  -  -  -  259 

Prob.  VII.  To  find  the  Sun’s  Longitude,  Semidiameter  and 
Hourly  Motion,  and  the  apparent  Obliquity  of  the 
Ecliptic,  for  a  given  time,  from  the  Tables,  -  262 

Prob.  VIII.  The  Obliquity  of  the  Ecliptic,  and  the  Sun’s 
Longitude  being  given,  to  find  the  Right  Ascension 
and  Declination,  -  264 

Prob.  IX.  Given  the  Obliquity  of  the  Ecliptic  and  the  Sun’s 

Right  Ascension,  to  find  the  Longitude  and  Declination,  265 
Prob.  X.  The  Obliquity  of  the  Ecliptic  and  the  Sun’s  Longi¬ 
tude  being  given,  to  find  the  angle  of  Position,  -  266 

Prob.  XI.  To  find  from  the  Tables,  the  Moon’s  Longitude, 

Latitude,  &c.  -  267 

Prob.  XII.  To  find  the  Moon’s  Longitude,  &c.  from  the 

Nautical  Almanac,  -  274 

Prob.  XIII.  To  find  the  Reductions  of  Parallax  and  Latitude,  278 
Prob.  XIV.  To  find  the  Mean  Right  Ascension  and  Decli¬ 
nation,  or  Longitude  and  Latitude  of  a  Star,  from  the 
Tables,  -  279 

Prob.  XV.  To  find  the  Aberrations  of  a  Star  in  Right  Ascen¬ 
sion  and  Declination,  -  -  -  -  280 

Prob.  XVI.  To  find  the  Nutations  of  a  Star  in  Right  Ascen¬ 
sion  and  Declination,  -  282 

Prob.  XVII.  To  find  the  Aberrations  of  a  Star  in  Longitude 

and  Latitude,  -----  284 

Prob.  XVIII.  To  find  the  Nutation  of  a  Body  in  Longitude,  285 

Prob.  XIX.  The  Obliquity  of  the  Ecliptic  and  the  Right 
Ascension  and  Declination  of  a  Body  being  given,  to 
find  the  Longitude  and  Latitude,  -  -  285 

Prob.  XX.  The  Obliquity  of  the  Ecliptic  and  the  Longitude 
and  Latitude  of  a  Body  being  given,  to  find  the  Right 
Ascension  and  Declination,  -  -  287 

Prob.  XXI.  The  Obliquity  of  the  Ecliptic  and  the  Longi¬ 
tude  and  Declination  of  a  Body  being  given,  to  find 
the  Angle  of  Position,  -  -  -  289 

Prob.  XXII.  To  find  the  Time  of  a  Star’s  Passage  over  the 

Meridian,  -  -  -  -  290 

Prob.  XXIII.  To  find  the  Time  of  the  Moon’s  Passage  over 

the  Meridian,  -  -  -  -  391 


CONTENTS. 


IX 


PAGE. 

Prob.  XXIV.  To  find  the  Time  of  the  Sun’s  Rising  and 

Setting,  -  -  *  -  -  292 

Prob.  XXV.  To  find  the  Time  of  the  Moon’s  Passage  over 

the  Meridian,  from  the  Nautical  Almanac,  -  293 

Prob.  XXVI.  To  find  the  Moon’s  Declination  from  the 

Nautical  Almanac,  -  -  -  294 

Pr  3B.  XXVII.  To  find  the  Time  of  the  Moon’s  Rising  or 

Setting,  -  -  -  -  296 

Prob.  XXVIII.  To  find  the  Longitude  and  Altitude  of  the 

Nonagesirnal,  -  29S 

Prob.  XXIX.  To  find  the  Moon’s  Parallax  in  Longitude  and 

Latitude,  -  -  -  -  - '  301 

Prob.  XXX.  To  find  the  Time  of  New  or  Full  Moon  by 

the  Tables,  -  309 

Prob.  XXXI.  To  find  the  Time  of  New  or  Full  Moon  by 

the  Nautical  Almanac,  ...  313 

Prob.  XXXII.  To  determine  what  Eclipses  maybe  expected 
to  occur  in  a  given  year,  and  the  Times  nearly  at 
which  they  will  take  place,  -  -  315 

Prob  XXXIII.  To  Calculate  an  Eclipse  of  the  Moon,  319 
Prob.  XXXIV.  To  Project  an  Eclipse  of  the  Moon,  -  324 

Prob.  XXXV  To  Project  an  Eclipse  of  the  Sun,  -  328 

Prob.  XXXVI.  To  Calculate  an  Eclipse  of  the  Sun,  337 
Prob.  XXXVII.  To  find  by  Projection  the  Latitudes  and 
Longitudes  of  the  Places  at  which  an  Eclipse  of  the 
Sun  is  Central,  for  different  times  during  the  continu¬ 
ance  of  the  Central  Eclipse,  -  345 

pRon  XXXVIII.  To  Project  an  Occultation  of  a  Fixed  Star 

by  the  Moon,  -----  348 

Prob.  XXXIX.  Given  the  Moon’s  true  Longitude,  to  find 
the  corresponding  time  at  Greenwich  by  the  Nautical 
Almanac,  -  354 

Prob.  XL.  To  find  the  Longitude  of  a  Place  from  the  ob- 
served  time  of  beginning  or  end  of  an  Occultation  of 
a  Fixed  Star  by  the  Moon,  -  355 


1 


The  following  Alphabet  is  given  in  order  to  facilitate,  to  the  stu¬ 
dent  who  is  unacquainted  with  it,  the  reading  of  those  parts  in 
which  the  Greek  letters  are  used: 


Letters. 

Names. 

A  CL 

Alpha 

B  /3£ 

Bela 

r  y  f 

Gamma 

A  J 

Delta 

E  t 

Epsilon 

z  U 

Zeta 

H  v, 

Eta 

©  3-0 

Theta 

I  i 

Iota 

K  K 

Kappa 

A  A 

Lambda 

M  /LL 

Mu 

N  v 

Nu 

s  1 

Xi 

O  o 

Omicron 

n  zttt 

Pi 

p  f  f 

liho 

2  <r  ? 

Si<;ma 

T 

Tau 

TO 

Upsilon 

<P<f> 

Phi 

x* 

Chi 

^  4, 

Psi 

fl  0> 

Omega 

AN 


ELEMENTARY  TREATISE 

ON 


ASTRONOMY. 

PART  I. 

CHAPTER  I. 

General  Phenomena  of  the  Heavens. 

1.  Astronomy  is  the  science  which  treats  of  the  ap¬ 
pearances,  motions,  distances,  and  magnitudes  of  the 
heavenly  bodies.  That  part  of  the  science  in  which  the 
causes  of  their  motions  are  considered,  is  called  P/iy- 
sical  Astronomy. 

2.  If,  in  a  clear  night,  we  fix  our  attention  on  the 
heavens,  and  make  continued  or  repeated  observations 
on  the  stars,  we  shall  find  that  they  retain  the  same 
situations  with  respect  to  each  other,  but  that  with  re¬ 
spect  to  the  earth  they  undergo  a  continual  change. 
Those  to  the  eastward  will  be  seen  to  ascend,  and 
others  will  come  into  view  or  rise ;  those  to  the  west¬ 
ward  will  be  seen  to  descend  and  will  go  out  of  view, 
or  set. 

3.  If  we  direct  our  attention  to  the  north,  different 
phenomena  will  present  themselves.  Many  stars  will 

2 


2 


ASTRONOMY. 


be  seen  that  do  not  set.*  They  appear  to  revolve  or 
describe  circles  about  a  certain  star,  that  seems  to  re¬ 
main  stationary:  this  stationary  star  is  called  the  Volar 
Star.  Those  stars  that  do  not  set,  are  called  Circum¬ 
polar  Stars. 

4.  When  the  polar  star  is  accurately  observed,  it 
ceases  to  appear  stationary,  and  is  found  to  have  an 
apparent  motion  in  a  small  circle,  about  a  point  from 
which  the  different  parts  of  the  circumference  are 
equally  distant.  This  point  is  called  the  North  Pole. 
It  is  in  reality  about  this  point,  and  not  the  polar  star, 
that  the  apparent  revolutions  of  the  stars  are  performed. 

5.  The  stars  appear  to  move,  from  east  to  west,  ex¬ 
actly  as  if  attached  to  the  concave  surface  of  a  hollow 
sphere  which  revolves  on  its  axis  in  a  space  of  time, 
nearly  equal  to  24  hours.  This  motion,  which  is  com¬ 
mon  to  all  the  heavenly  bodies,  is  called  the  Diurnal 
Motion. 

6.  If  wre  examine  the  situations  of  the  Moon,  on  suc¬ 
cessive  nights,  we  shall  find  that  she  changes  her  posi¬ 
tion  among  the  stars,  and  advances  from  west  to  east. 

7.  The  Sun  also  appears  to  partake  of  this  motion, 
relative  to  the  stars.  This  may  be  inferred  from  ob¬ 
serving  the  stars  in  the  west  after  the  sun  has  set.  If 
our  observations  be  continued  for  a  number  of  succes¬ 
sive  evenings,  we  shall  find  that  the  sun  continually 
approaches  to  the  stars,  situated  to  the  eastward  of  him. 

8.  Besides  the  sun  aud  moon,  there  are  ten  stars 
which  change  their  situations,  with  respect  to  the  other 
stars,  and  have  a  motion  among  them.  These  are  called 
Planets.  Five  of  them,  named,  Mercury ,  Venus , 

*  Here,  and  in  other  parts  of  the  work,  unless  the  contrary  is  mentioned, 
the  Observer  is  supposed  to  be  in  the  United  States,  or  in  the  southern  or 
middle  parts  of  Europe. 


ASTRONOMY. 


3 


Mars ,  Jupiter,  and  Saturn,  are  visible  to  the  naked 
eye,  and  were  known  to  the  ancients.  The  other  five, 
named,  Vesta,  Juno,  Ceres ,  Pallas,  and  Uranus 
can  not  be  seen  w  ithout  the  aid  of  a  telescope,  and  have 
not  been  long  known. 

The  stars  which  do  not  sensibly  change  their  situa¬ 
tions,  with  respect  to  one  another,  are  called  Fixed 
Stars. 

9.  There  are  some  stars,  that  occasionally  appear  in 
the  heavens,  which  have  a  motion  among  the  fixed  stars, 
and  only  continue  visible  for  a  few  weeks  or  months. 
They  are  commonly  accompanied  by  a  faint  brush  of 
light,  called  a  tail.  These  are  named  Comets. 

10.  If  a  person,  placed  on  the  margin  of  the  sea,  ob¬ 
serve  a  vessel  receding  from  the  land,  he  will  first  lose 
sight  of  the  hull,  then  of  the  lower  parts  of  the  sails, 
and  lastly  of  the  topsails.  This  will  be  the  case,  in 
whatever  direction  the  vessel  pursues  her  course,  or  in 
whatever  part  of  the  earth  the  observation  is  made.  We 
lienee  conclude  that  the  surface  of  the  sea  is  convex.  It 
is  also  well  known,  that  vessels  have  sailed  entirely 
round  the  earth,  in  different  directions.  From  these 
circumstances,  it  is  inferred  that  the  form  of  the  earth 
is  globular. 

11.  In  astronomical  investigations,  except  when 
great  accuracy  is  required,  it  is  usual  to  consider  the 
earth  as  a  sphere. 

12.  The  angular  distance  between  any  two  of  the 
fixed  stars,  is  found  to  be  the  same,  in  whatever  part 
of  the  earth’s  surface  the  observation  is  made.  It  fol¬ 
lows,  therefore,  that  the  distance  of  the  stars  from  the 

*  The  planet  Uranus,  which  was  discovered  by  Dr.  Herschel,  was  by  him 
named  Georgian  Sidus,  in  honour  of  his  patron,  King  George  III.  By  the 
French  it  was  called  Herschel.  It  is  now  generally  known  by  the  name 
given  in  the  text 


ASTRONOMY. 


4 

earth  is  so  great,  that  the  earth’s  diameter,  compared 
with  it,  is  insensible. 

13.  It  is  not  supposed  that  the  fixed  stars  are  all  at 
the  same  distance  from  the  earth.  But  since  their  dis¬ 
tances  are  so  immensely  great  that  the  most  accurate 
observations  do  not  indicate  a  difference,  they  are  con¬ 
sidered  as  placed  in  the  concave  surface  of  a  sphere, 
having  the  same  centre  with  the  earth. 


CHAPTER  II. 

Definitions  of  Terms . — Astronomical  Instruments . 

1.  The  straight  line  which  passes  through  the  North 
Pole,  and  through  the  centre  of  the  earth,  is  called  the 
Axis  of  the  Heavens .  It  is  the  line  about  which  the 
heavens  appear  to  revolve. 

2  The  point  in  which  the  axis  of  the  heavens  meets 
the  southern  part  of  the  celestial  sphere,  is  called  the 
South  Pole . 

3.  The  points  in  which  the  axis  of  the  heavens  in¬ 
tersects  the  surface  of  the  earth,  are  called  the  J\T orth 
and  South  Poles  of  the  Earth . 

4.  A  plane  which  passes  through  the  centre  of  the 
earth,  and  is  perpendicular  to  the  axis  of  the  heavens, 
intersects  the  celestial  sphere  in  a  circle,  which  is  called 
the  Celestial  Equator ,  or  simply  the  Equator.  The 
circle  in  which  this  plane  cuts  the  surface  of  the  earth, 
is  called  the  Terrestrial  Equator. 

5.  If  at  any  place  on  the  earth’s  surface  a  straight 
line  in  the  direction  of  gravity,  that  is  in  the  direction 
of  the  plumb  line,  when  a  plummet  is  freely  suspended 
and  is  at  rest,  be  produced  upward,  the  point  in  which 
it  cuts  the  celestial  sphere,  is  called  the  Zenith  of  the 


ASTRONOMY. 


5 


place.  If  tbe  same  line  be  produced  downward,  the 
point  in  which  it  cuts  the  opposite  part  of  the  sphere, 
is  called  the  Nadir . 

6.  A  plane  which  passes  through  any  place,  and  is 
perpendicular  to  the  straight  line  joining  the  zenith  and 
nadir,  cuts  the  celestial  sphere  in  a  circle,  which  is 
called  the  Horizon ,  or,  sometimes,  the  Sensible  Hori¬ 
zon. 

The  circle  in  which  a  plane,  passing  through  the 
centre  of  the  earth,  and  parallel  to  the  horizon,  cuts  the 
celestial  sphere,  is  called  the  Rational  Horizon . 

7.  A  great  circle  which  passes  through  the  poles  of 
the  heavens  and  through  the  zenith  of  a  place,  is  called 
the  Meridian  of  that  place. 

The  meridian  cuts  the  horizon  at  right  angles,  in  two 
points,  called  the  N orth  and  South  Points  of  the  Ho¬ 
rizon . 

8.  The  intersection  of  the  plane  of  the  meridian  with 
the  earth’s  surface,  is  called  the  Meridian  Line ,  or 
Terrestrial  Meridian. 

9.  The  arc  of  the  meridian  intercepted  between  the 
zenith  and  equator,  is  called  the  Latitude  of  the  place. 

10.  Circles,  which  pass  through  the  zenith  and  na¬ 
dir  of  any  place,  are  called  Vertical  Circles ,  and  are 
perpendicular  to  the  horizon  of  the  place. 

11.  The  vertical  circle  which  is  at  right  angles  to 
the  meridian,  is  called  the  Prime  Vertical. 

The  prime  vertical  intersects  the  horizon  in  two 
points,  called  the  East  and  West  Points  of  the  Horizon. 

12.  The  arc  of  a  vertical  circle,  intercepted  between 
a  star  and  the  horizon,  is  called  the  Altitude  of  the  star; 
and  the  arc  of  the  horizon,  intercepted  between  the 
said  vertical  and  the  meridian,  is  called  the  Azimuth 
of  the  star. 


6 


ASTRONOMY. 


The  definitions  of  other  astronomical  terms  will  be 
found  in  succeeding  parts  of  the  work,  when  such  know¬ 
ledge  of  the  subject  shall  have  been  obtained,  as  will 
render  them  easily  understood. 

13.  For  making  astronomical  observations  various 
instruments  are  used,  some  of  which,  with  the  purposes 
to  which  they  are  applied,  it  will  be  proper  briefly  to 
mention. 

14.  The  Astronomical  Quadrant  is  an  instrument 
used  to  take  the  altitude  of  a  heavenly  body.  It  is  made 
of  different  sizes,  but  generally  of  two,  three,  or  more 
feet  radius.  The  quadrantal  arc  or  limb  is  divided  into 
90  equal  parts  or  degrees,  and  these  degrees  are  sub¬ 
divided  into  smaller  parts,  according  to  the  size  of  the 
instrument.  To  the  quadrant  a  telescope  is  attached, 
having  a  motion  about  the  centre  of  the  quadrant  and 
carrying  with  it  a  vernier  index  that  moves  along  the 
graduated  limb  and  increases  its  subdivisions,  which, 
by  this  means,  is  generally  extended  to  seconds.  In  the 
eye  tube  of  the  telescope  a  ring  is  placed,  having  two 
very  fine  wires  attached  to  it,  crossing  each  other  at 
right  angles  in  the  centre.  The  intersection  of  these 
wires  is  made  to  coincide  accurately  with  the  focus  of 
the  eye  glass,  and  serves  to  determine  the  line  of  sight, 
or,  as  it  is  technically  called,  the  Line  of  Collimation 
of  the  telescope. 

1 5.  The  Astronomical  Circle  is  an  instrument  by 
which  an  observer  may,  at  the  same  time,  obtain  the 
altitude  and  azimuth  of  a  heavenly  body.  It  has  two 
graduated  circles,  one  horizontal  for  the  azimuth,  and 
the  other  vertical  for  the  altitude.  A  telescope  is  fixed 
to  the  vertical  circle,  and  moves  with  it.  Most  astro¬ 
nomical  observations  may  be  accurately  made  with  an 
instrument  of  this  kind. 


ASTRONOMY. 


7 


16.  A  Transit  Instrument  is  a  telescope  fitted  up 
in  such  a  manner,  that  its  line  of  collimation  may  be 
made  to  move  accurately  in  the  plane  of  the  meridian. 
It  is  used  for  observing  the  passage  of  a  heavenly  body 
over  the  meridian. 

17-  A  Micrometer  is  an  instrument  attached  to  tele¬ 
scopes,  by  means  of  which  small  angles  may  be  mea¬ 
sured  with  an  extreme  degree  of  precision. 

18.  The  Astronomical  Clock  is  not  very  different 
from  the  common  clock.  Its  pendulum  rod  is  so  con¬ 
structed,  that  its  length  is  not  sensibly  affected  by 
changes  in  the  temperature  of  the  air.  The  hours  on  the 
face  are  marked  from  1  to  21. 

The  student  who  wishes  to  see  particular  descrip¬ 
tions  of  astronomical  instruments,  accompanied  by  en¬ 
gravings,  may  consult  Vince’s  Practical  Astronomy , 
Traite  I)’ Astronomie 'par  Delambre ,  or  Rees’s  Cyclo¬ 
pedia. 

CHAPTER  III. 

Meridian  Line. — Sidereal  Bay. — Biurnal  Motion . — 
Refraction. 

1.  Let  Z  R,  Fig.  1.  represent  the  northern  part  of 
the  meridian  of  a  place,  Z  the  zenith,  P  the  pole,  II 
RO  the  horizon,  SS'Gthe  circle  which  one  of  the^fixed 
stars  appears  to  describe  in  its  diurnal  motion,  and  S 
and  S'  different  situations  of  the  star,  the  former  in  the 
eastern  and  the  latter  in  the  western  part  of  the  heavens: 
also  let  PS  and  PS'  be  arcs  of  declination  circles, 
ZSA  and  ZS'B  arcs  of  vertical  circles,  and  let  the 
situations  S  and  S'  of  the  star  in  its  apparent  circle  be 
such  that  the  altitudes  AS  and  BS'  are  equal 


ASTRONOMY. 


Then  if,  as  it  appears  to  do,  the  star  continues  at  the 
same  distance  from  the  pole  P,  the  arc  PS  =  PS';  also 
because  ZA  ==  ZB,  being  each  quadrants,  and  AS  = 
BS',  we  have  ZS  =  ZS',  and  PZ  is  common  to  the 
two  triangles  PZS  and  PZS';  therefore  the  angle 
PZS  =  PZS',  and  the  arc  AR  =  BR,  these  arcs  be¬ 
ing  the  measures  of  the  angles  PZS  and  PZS'.  Now, 
RA  and  RB  are  the  azimuths  of  the  star  when  in  the 
situations  S  and  S'  (2.  12.)*  If  therefore  the  altitude 
and  bearing  of  a  star  be  observed  when  in  the  eastern 
part  of  the  heavens,  and  if  its  bearing  be  again  observ¬ 
ed,  when  it  arrives  at  the  same  altitude  in  the  western 
part  of  the  heavens,  the  line  bisecting  the  angle  made 
by  these  bearings  will  be  a  meridian  line,  (2.  8.) 

2.  In  conformity  with  appearance,  we  have,  in  the 
preceding  article,  made  the  assumption,  that  the  appa¬ 
rent  diurnal  motion  of  a  star  is  in  a  circle.  The  proba¬ 
bility  of  this  assumption  being  true,  is  increased  by  the 
fact  that  repeated  accurate  observations  on  the  same 
star  with  different  altitudes,  or  similar  observations  on 
any  other  star,  give  the  same  situation  for  the  meridian 
line. 

3.  When  an  accurate  meridian  line  has  been  thus 
obtained,  an  astronomical  circle,  or  a  transit  instrument, 
may  be  so  adjusted,  that  the  line  of  colli mation  of  its 
telescope,  shall  move  in  the  plane  of  the  meridian. 

4.  When  by  a  good  clock  the  exact  time  is  observed 
from  the  time  of  a  fixed  star  passing  the  meridian  on 
any  evening  to  the  time  of  its  passage  on  the  next  eve¬ 
ning,  and  this  observation  is  repeated  on  several  suc¬ 
cessive  evenings,  it  is  found  that  the  interval  of  time 

*  The  first  number  refers  to  the  chapter,  and  the  second  to  the  article. 
When  a  reference  is  mad%to  an  article  in  the  same  chapter,  the  number  of 
the  article  only  i9  given. 


CHAPTER  III, 


9 


between  its  passages  on  any  two  succeedirig  evenings 
is  the  same.  Similar  observations  on  different  stars  give 
the  same  interval  of  time.  This  is  true,  not  only  for  the 
time  between  two  successive  passages  of  a  star  over  the 
meridian,  but  also  for  the  time  from  a  star  being  at  any 
altitude  to  its  return  to  the  same  altitude  on  the  suc¬ 
ceeding  evening.  It  appears  therefore  very  probable 
that  the  diurnal  motion  of  a  star  is  uniform. 

5.  The  time  between  two  successive  passages  of  a 
star  over  the  meridian  is  called  a  Sidereal  Day.  And 
a  clock  that  is  so  regulated  as  to  move  through  24 
hours  in  the  course  of  a  sidereal  day  is  said  to  be  regu- 
lated  to  Sidereal  Time. 

6.  We  have  inferred  as  probable,  that  the  diurnal 
motion  of  a  star  is  performed  in  a  circle  about  the  pole 
of  the  heavens,  and  that  its  motion  in  that  circle  is  uni¬ 
form.  If  this  is  the  case  it  is  evident  that  the  distance 
PS'  is  constant,  and  that  the  angle  ZPS  must  in* 
crease  uniformly  with  the  time.  As  the  star  moves 
through  the  whole  circle  or  3ff0°  in  24  sidereal  hours, 
it  must,  if  its  motion  is  uniform,  move  through  15°  in 
each  hour,  and  consequently  the  angle  ZPS  must  in¬ 
crease  at  the  rate  of  15°  per  hour.  Now,  if  PZ,  the 
distance  of  the  pole  from  the  zenith,  and  PS',  the  dis¬ 
tance  of  a  star  from  the  pole,  are  known,  and  if  the  al¬ 
titude  BS ,  or  its  complement,  the  zenith  distance  ZS' 
of  the  star,  be  observed,  the  angle  ZPS'  may  be  calcu¬ 
lated.  Observations  and  calculations  thus  made  on  a 
star  at  different  times  during  the  same  night,  prove  that 
the  angle  ZPS  varies  as  the  time;  and,  therefore,  that 
the  diurnal  motion  of  a  star  is  uniform . 

7.  In  the  preceding  article  the  distances  PZ  and  PS' 
are  supposed  to  be  known.  A  method  of  obtaining 
them  is  now  to  be  explained.  Let  F  and  G  be  the  situ 

3 


10 


ASTRONOMY. 


ations  of  tlie  star  on  the  meridian  above  and  below  the 
pole.  Then  we  have 

ZF  =  PZ  —  PF 
ZGr  =  PZ  +  PGr  =  PZ  +  PF 
therefore  ZB  —  ZF  =  %  PF,  or  PF  =  \  (ZGr —  ZF) 
also  ZGr  +  ZF  =  2  PZ,  or  PZ  =  f  (ZG  +  ZF) 

Let  K  and  L  be  the  situations  on  the  meridian  of 
another  star  at  a  different  distance  from  the  pole.  Then 
we  have  in  like  manner  PZ  =  \  (ZK  +  ZL.)  But 
observations  made  on  different  stars  at  different  dis¬ 
tances  from  the  pole  do  not  give  the  same  result  for 
PZ.  It  is  found  that  the  value  of  PZ  thus  obtained  is 
less,  as  the  distance  of  the  star  from  the  pole  is  greater; 
that  is,  as  it  is  nearer  to  the  horizon  when  on  the  meri¬ 
dian  below  the  pole.  When  the  altitude  RP  of  the 
pole  is  40  or  50°,  and  one  of  the  stars  observed  is  the 
polar  star,  and  the  other  is  one  that  at  its  passage  of 
the  meridian  below  the  pole,  is  very  near  the  horizon, 
the  difference  between  the  values  of  PZ  obtained  from 
them,  amounts  to  about  half  a  degree. 

This  effect  is  produced  by  the  action  of  the  earth’s 
atmosphere  on  the  rays  of  light  from  the  stars,  and  is 
called  Atmospherical  Refraction ,  and  sometimes  As¬ 
tronomical  Refraction . 

OF  REFRACTION. 

8.  It  is  known  that  when  a  ray  of  light  passes  ob¬ 
liquely  from  one  medium  to  another  of  different  density, 
its  direction  is  changed. 

Let  FHGB  Fig  2,  be  a  vertical  section  of  a  vessel  whose  sides 
are  opaque.  An  object  placed  on,  the  bottom  at  E  could  not,  when 
the  vessel  is  empty,  be  seen  by  an  eye  placed  at  0.  But  if  the 


CHAPTER  III. 


11 


vessel  be  filled  with  water,  the  object  will  become  visible  in  the 
direction  OB,  and  will  appear  as  though  it  were  really  at  D.  A 
ray  of  light,  therefore,  which  passes  obliquely  from  water  to  air  is 
refracted  so  as  to  make  a  greater  angle  with  the  perpendicular  to 
the  common  surface,  than  if  it  passed  on,  without  suffering  a 
change  in  its  direction.  Again,  a  ray  of  light  passing  from  an 
object  at  0,  in  the  direction  OB  will,  when  the  vessel  is  empty, 
meet  the  bottom  in  D.  But  if  the  vessel  be  filled  with  water,  the 
ray  of  light  will  be  refracted  on  entering  the  water,  and  will  take 
the  direction  BE,  so  that  to  an  eye  at  E,  it  will  appear  to  come 
from  the  point  A,  and  therefore  the  object  will  appear  to  be  more 
elevated  than  it  really  is.  The  same  effects,  though  different  in 
degree,  take  place  when  a  ray  of  light  passes  from  air  into  a  va¬ 
cuum,  or  the  contrary. 

9.  The  angle  contained  between  the  directions  of  the  , 
direct  and  refracted  rays,  is  called  the  Jingle  of  Re¬ 
fraction,  or  simply  the  Refraction . 

10.  It  is  found  by  experiment,  that  for  the  same  two 
mediums,  except  when  the  ray  of  light  passes  very  ob¬ 
liquely  from  one  to  the  other,  the  sine  of  the  angle  con¬ 
tained  between  the  direct  ray  and  the  perpendicular  to 
the  common  surface,  is  equal  to  the  sine  of  the  angle 
contained  between  the  refracted  ray  and  the  same  per¬ 
pendicular,  multiplied  by  some  constant  quantity. 

If  ZB  be  perpendicular  to  the  common  surface  FB,  of  the  two 
mediums,  and  OB  be  the  direct  ray  and  AB  the  direction  of  the 
refracted  ray,  we  have  m  sin  ZBA  =  sin  ZBO  ==  sin  (ZBA  -{- 
ABO.)  The  value  of  m  is  constant  for  the  same  two  mediums, 
but  is  greater  or  less,  according  as  the  difference  of  density  of  the 
mediums,  is  greater  or  less. 

11.  The  atmosphere  extends  to  the  height  of  some 
miles,  and  its  surface  is  supposed  to  be  nearly  concen¬ 
tric  with  the  surface  of  the  earth.  It  has  been  found 
by  experiments,  in  ascending  high  mountains,  that  its 


12 


ASTRONOMY. 


density  gradually  decreases,  with  an  increase  of  dis¬ 
tance  from  the  general  surface  of  the  earth.  Hence  a 
ray  of  light  which  enters  it  obliquely,  passing  continu¬ 
ally  from  a  rarer  to  a  denser  medium,  has  its  direction 
continually  changed,  and  its  path  will  therefore  be  a 
curve,  concave  towards  the  earth.  This  curve  coincides 
With  a  vertical  plane,  because  as  the  density  of  the  at¬ 
mosphere,  on  each  side  of  such  plane,  is  the  same,  there 
is  no  cause  for  its  deviating  either  way.  Refraction, 
therefore ,  makes  the  apparent  altitude  of  a  star ,  great¬ 
er  than  the  true ;  but  it  does  not  change  its  azimuth . 

The  curvilinear  path  of  a  ray  of  light,  passing 
through  the  atmosphere,  differs  but  little  from  a  right 
line,  except  near  the  horizon,  where,  on  account  of  its 
greater  obliquity,  its  direction  undergoes  a  greater 
change. 

12.  The  refraction,  except  near  the  horizon,  varies, 
nearly  as  the  tangent  of  the  apparent  zenith  distance. 


Let  N  represent  the  apparent  zenith  distance  of  a  star,  and  r, 
the  refraction  corresponding.  Then  N  4  r  =  the  true  zenith 
distance,  and,  therefore,  (10.) 

m  sin  N  ==  sin  (N  4  r)  =  sin  N  cos  r  4.  cos  N  sin  r  (App.*  13.) 


cos  iN 

or  m  =  cos  r  4-  — — ^  sin  r  =  cos  r  4-  cot  N  sin  r. 

TsinN 

But  as  r  is  small,  cos  r  —  1,  and  sin  r  =  r,  nearly.  Therefore 


m  =  1  -f  r  cot  N,  or  r  == 


cot  N 


=  (m  r —  1 )  tan  N. 


13.  It  is  evident  that  the  refraction  can  not  vary  ac¬ 
curately  as  the  tangent  of  the  apparent  zenith  distance, 
because  in  that  case,  when  the  zenith  distance  is  90°, 
it  w  ould  be  infinite.  And  in  all  cases,  w  hen  the  alti¬ 
tude  is  small,  and  consequently  when  the  rays  of  light 


*  Appendix  at  the  end  of  part  1st, 


CHAPTER  III.  13 

enter  tlie  atmosphere  very  obliquely,  it  would  be  too 
great. 

Dr.  Bradley  found  that  the  refraction  is  more  nearly 
equal  to  the  product  of  the  tangent  of  the  difference 
between  the  apparent  zenith  distance  and  three  times 
the  refraction,  multiplied  by  a  constant  quantity. 

His  formula  is,  r—  57"  tan  (N — 3r);  in  which  57"  is  the 
refraction  at  45°  apparent  zenith  distance, 

14.  From  the  observed  altitudes  or  zenith  distances 
of  two  circumpolar  stars,  when  on  the  meridian,  both 
above  and  below  the  pole,  the  true  distance  of  the 
pole  from  the  zenith,  the  refraction  for  each  meri¬ 
dian  altitude  of  the  stars,  and  the  true  polar  distances 
of  the  stars,  may  be  obtained. 


In  Fig.  1 .  let  P  be  the  true  place  of  the  pole,  G  and  F  the 
true  situations  of  a  star  on  the  meridian,  below  and  above  the 
pole,  and  L  and  K,  those  of  another  star.  Also  let  N  and  n  re¬ 
present  the  apparent  zenith  distances  of  the  star  at  G  and  F,  R 
and  r  the  corresponding  refractions,  N'  and  n'  the  apparent  ze¬ 
nith  distances  of  the  other  star  at  L  and  K,  and  R'  and  /,  the 
corresponding  refractions.  Then, 

ZG  =  N-pR,  ZF  =  tt-f  r,  ZL  =  N'-j-R',  and ZK  =  n'-p  r'; 
therefore  (7),  PZ  ==  ~  (N-pR+n-pr)  =  -J  (N'-f  R'-pn'+r',) 
and  N-pR-pn-pr  =  N'-pR'-pn'-pr', 
or  R-pr — R' — r'  =  N'-pn' — N — n. 


Now  supposing  the  refraction  to  vary  as  the  tangent  of  the  ap¬ 
parent  zenith  distance,  (12)  we  have, 


r  == 


m —  1  = 
R  tan  n 


R 


R 


r' 


tan  N  tan  n 
R  tan  N' 


R+ 


tan  N 
R  tan  n 


R'  = 


tan  N 


tan  N 
R  tan  N' 
tan  N 


tan  N' 
and  r'  -= 


tan  n' 
R  tan  n' 
tan  N 


whence, 

therefore, 


— an  -  =  N'+n'—  N 
tan  N 


n. 


14} 


ASTRONOMY. 


From  which  we  obtain  R  =  — W+n'—X-n)  tan  N 

tan  N-ftan  n — tan  N  — tan  n 

Whence  r,  R  and  r  become  knoAvn,  and  consequently  PZ  =  \ 
(N-j-w-f  R-frj;  also  the  polar  distance  PF  =  ZP —  ZF  = 
ZP  —  (»+r),  and  PK  =  ZP  — ZK  =  ZP  — (n'  +  r). 

This  method  of  finding  the  refraction  is  by  Boscovich.  When 
neither  of  the  zenith  distances  exceeds  70°  or  75°,  it  gives  it  with 
considerable  accuracy. 

Id.  When  the  true  distance  of  the  pole  from  the 
zenith,  and  the  apparent  zenith  distance  and  corres¬ 
ponding  refraction,  of  a  star  on  the  meridian,  are 
known,  the  true  polar  distance  of  any  other  star  may 
he  determined  from  its  observed  meridian  altitude  or 
zenith  distance. 

If  M  be  the  true  situation  of  the  star  on  the  meridian,  and  we 
put  n"  for  the  observed  apparent  zenith  distance,  and  r"  for  the 
corresponding  refraction,  we  have, 

tan  n  tan  n"  _  r  tan  n" . 

_  = - or  r  =  - ; 

r  r"  tan  n 

consequently  PM  =  PZ-f  ZM  =  ZP  +  n"  4-  r",  becomes  known-. 

16.  The  true  zenith  distance  of  the  pole  and  polar 
distance  of  a  star  being  known,  the  refraction  may  be 
found  for  any  observed  altitude,  by  knowing  also  the 
interval  between  the  times  of  observation  and  of  the 
star’s  passage  over  the  meridian. 

Let  S"  be  the  place  of  the  star.  Then  in  the  triangle  ZPS", 
the  sides  ZP  and  PS"  are  given,  and  also  from  the  observed  in¬ 
terval  of  time,  the  angle  ZPS".  Consequently  the  side  ZS"  may 
be  calculated  The  difference  between  ZS"  thus  obtained,  and 
the  observed  zenith  distance,  is  the  corresponding  refraction.  In 
this  way  the  refraction  may  be  found  for  different  apparent  alti¬ 
tudes,  from  the  horizon  to  the  zenith. 

17.  The  refraction  at  a  given  altitude  is  subject  to 


CHAPTER  III. 


15 


some  change  depending  on  the  variation  in  the  state  of 
the  air  as  indicated  by  the  barometer  and  thermometer. 
The  refractions  which  have  place  when  the  barometer 
stands  at  29.6  inches  and  the  thermometer  at  50°  are 
called  mean  refractions.  Table  II.  contains  the  mean 
refractions  for  different  altitudes  from  the  horizon  to 
the  zenith.  Above  the  altitude  of  15  and  20°,  these 
are  at  all  times  sufficiently  accurate,  except  in  cases 
when  the  greatest  precision  is  required. 

For  mule  have  been  investigated  for  obtaining  the 
refraction  with  reference  to  the  state  of  the  barometer 
and  thermometer;  but  these  investigations  do  not  be¬ 
long  to  an  elementary  treatise. 

OTHER  EFFECTS  OF  REFRACTION. 

18.  As  refraction  elevates  the  heavenly  bodies  in 
verticle  circles,  and  as  these  circles  continually  ap¬ 
proach  each  other  from  the  horizon  till  they  meet  in 
the  zenith,  it  is  evident  that  the  apparent  distance  of 
any  two  of  those  bodies  must  be  less  than  the  true  dis¬ 
tance. 

19.  The  refraction  increases  with  an  increase  of 
zenith  distance.  The  lower  part  of  the  sun  or  moon 
is  therefore  more  refracted  than  the  upper  part,  so  that 
the  vertical  diameter  is  shortened  and  the  body  ap¬ 
pears  of  an  elliptical  form.  This  effect  is  most  ob¬ 
servable  near  the  horizon,  where,  on  account  of  the 
more  rapid  increase  of  the  refraction,  the  difference  be¬ 
tween  the  vertical  and  horizontal  diameters  may 
amount  to  I  part  of  the  whole  diameter.  The  hori¬ 
zontal  diameter  also  suffers  a  slight  diminution.  (18) 

20.  At  the  true  horizon  the  refraction  is  about  38' f. 
Hence  it  follows  that  when  any  of  the  heavenly  bodies 
are  really  in  the  horizon,  they  appear  to  be  38' |  above 


16 


ASTRONOMY. 


it,  and  that  therefore  refraction  retards  their  setting, 
and  accelerates  their  rising. 

21.  When  the  sun  descends  below  the  horizon  of 
any  place,  its  rays  continue  for  some  time  to  reach  the 
upper  parts  of  the  atmosphere,  and  are  refracted  and 
reflected  so  as  to  occasion  considerable  light,  which 
gradually  diminishes  as  the  sun  descends  farther  be¬ 
low  the  horizon,  and  prevents  an  immediate  transition 
from  the  light  of  day  to  the  darkness  of  night.  The 
same  effect,  though  in  a  reverse  order,  takes  place  in 
the  morning  previous  to  the  sun’s  rising.  The  light 
thus  produced  is  called  the  Crejiusculum  or  Twilight. 


CHAPTER  IV. 

Latitude  of  a  Place. — Figure  and  Extent  of  the 
Earth . — Longitude. 

1.  Let  HZRN,  Fig.  9,  represent  the  meridian,  Pp 
the  axis  of  the  heavens,  Z  the  zenith,  HOR  the  hori¬ 
zon,  and  EOQ  the  equator,  the  latter  two  seen  edge-* 
wise.  Then  ZQ  is  the  latitude  of  the  place(2.9).  But 
ZQ  =  PQ— PZ  =  ZH  —  PZ  =  PH;  therefore  the  la¬ 
titude  of  a  place  is  equal  to  the  altitude  of  the  pole  at  that 
place.  A  method  of  obtaining  the  altitude  of  the  pole 
has  been  shown  (3.14). 

2.  Because  ZR  =  90°,  RQ  =  90°  —  ZQ;  therefore 
the  altitude  of  the  point  of  the  equator  which  is  on  the 
meridian,  at  any  time  is  equal  to  the  complement  of 
the  latitude  of  the  place. 

FIGURE  AND  EXTENT  OF  THE  EARTH. 

3.  By  the  figure  of  the  earth  is  meant  the  general 
form  of  its  surface,  supposing  it  to  be  smooth,  or  that 


CHAPTER.  IV. 


17 

the  surface  of  the  land  corresponded  with  the  surface 
of  the  ocean.  This  excludes  the  consideration  of  the 
irregularities  in  its  surface,  occasioned  by  mountains 
and  vallies,  which  indeed  are  very  minute  when  com¬ 
pared  with  the  whole  extent  of  the  earth, 
v  4.  Experiment  proves  that  the  direction  of  gravity, 
at  any  place  on  the  earth,  is  perpendicular  to  the  free 
surface  of  still  water:  Hence  it  is  perpendicular  to  the 
general  surface  of  the  earth  at  that  place.  The  straight 
line  which  represents  the  direction  of  gravity,  at  any 
place,  is  called  the  Vertical. 

5.  Let  EPQ  p,  Fig.  3,  be  a  meridian  of  the  earth, 
Vp  the  axis,  EQ  a  diameter  of  the  equator,  and  A  and 
B  two  places  on  the  meridian.  If  the  earth  be  a  sphere, 
the  direction  of  gravity  at  each  of  the  places  will  pass 
through  the  centre  of  the  earth,  and  therefore  the  angle 
EGA  will  be  the  latitude  of  A,  and  ECB  the  latitude 
of  B  (2.9).  Hence  if  the  latitudes  of  A  and  B  be  de¬ 
termined  (1),  the  angle  ACB  =  ECB  —  EGA,  be¬ 
comes  known.  The  distance  AB  may  be  obtained  by 
actual  measurement.  Then  as  the  angle  ACB  :  360° 
:  :  distance  AB  :  to  the  circumference  of  the  earth. 

6.  As  the  angle  ACB  :  1°  :  :  distance  AB  :  the 
length  of  a  degree  of  latitude.  Now,  if  the  earth  be  a 
sphere,  the  length  of  a  degree  of  latitude  must  be  the 
same  on  any  part  of  the  meridian.  But  it  has  been 
found  by  observation  and  measurement,  at  different 
places,  that  the  length  of  a  degree  increases  in  going 
from  the  equator  towards  the  pole.  At  the  equator 
the  length  of  a  degree  is  68m.  1280yds.  at  latitude  45° 
it  is  69m.  79yds.  and  at  latitude  66°|  it  is  69m. 
465  yds. 

7-  The  increase  in  the  length  of  a  degree  of  latitude, 
as  the  latitude  itself  increases,  proves  that  the  meridian 

4 


1,8 


ASTRONOMY. 


is  not  a  circle,  and  leads  to  the  supposition  that  it  is  an 
ellipse,  having  the  axis  of  the  earth  for  its  shorter  axis. 

Let  the  ellipse  EPQp,  Fig.  4,  represent  the  meri¬ 
dian,  P/?  the  axis  of  the  earth  aud  EQ  a  diameter  of 
the  equator.  Also  let  AD  be  perpendicular  to  the 
curve  at  A,  and  let  the  situation  of  the  point  B  be  such, 
that  BF,  perpendicular  to  the  curve  at  B,  may  make 
the  angle  BFP  =  EDA.  Then  the  angle  EDA  is 
the  difference  of  latitude  between  the  places  E  and  A, 
and  the  angle  BKP  is  the  difference  of  latitude  between 
the  places  B  and  P.  Now  it  is  evident  from  inspec¬ 
tion  of  the  figure  or  from  the  consideration  of  the  de¬ 
crease  of  curvature  from  E  to  P,  that  the  distance  BP, 
corresponding  to  a  given  difference  of  latitude  near  the 
pole,  is  greater  than  the  distance  EA,  corresponding 
to  the  same  difference  of  latitude,  near  the  equator. 

8.  Analytical  investigations  founded  on  the  measure 
of  a  degree  in  different  latitudes  and  on  different  meri¬ 
dians,  prove  that  a  meridian  is  nearly  an  ellipse,  and 
that  the  figure  of  the  earth  is  nearly  an  oblate  spheroid, 
having  the  polar  diameter,  to  the  equatorial  in  the  ratio 
pf  320  to  321. 

0,  Calculations,  made  from  the  most  accurate  mea¬ 
surements,  give  the  mean  diameter  of  the  earth  7920 
miles,  the  circumference  24880  miles,  and  the  length 
of  a  degree  of  a  great  circle  69  ot  miles.*  The  differ¬ 
ence  between  the  equatorial  and  polar  diameters,  is 
about  25  miles. 

10.  The  angle  contained  in  the  plane  of  the  meri¬ 
dian,  between  the  radius  of  the  equator  and  a  straight 
line  from  any  place  to  the  centre  of  the  earth,  is  called 

*  These  are  the  numbers,  nearly,  that  are  given  in  an  ingenious  essay  by 
R.  Adrain,  Prof,  of  Math,  in  Columbia  College,  and  published  in  the  Tran&, 
Actions  of  the  Araer.  Philos.  Society,  Vol.  1.  New  Series. 


CHAPTER  IV. 


19 


the  Reduced  Latitude  of  the  place.  And  the  angle 
contained  between  a  vertical  line  at  any  place  and  the 
straight  line  to  the  centre  of  the  earth,  is  called  the 
Reduction  of  Latitude .  Thus  the  angle  ECA  is  the 
reduced  latitude  of  the  place  A,  and  CAD  is  the  re- 
duction  of  latitude. 

Since  ECA  =  EDA  —  CAD,  it  is  plain  that  the 
Reduced  latitude  is  equal  to  the  true  latitude,  dimin¬ 
ished  by  the  reduction  of  latitude . 

11.  The  true  latitude  of  a  place  being  given,  the 
reduced  latitude  may  be  found  by  the  following  pro¬ 
portion.  The  square  of  the  equatorial  diameter ,  is  to 
the  square  of  the  polar  diameter ,  as  the  tangent  of  the 
true  latitude ,  is  to  the  tangent  of  the  reduced  latitude . 


Let  the  circle  EBQ,  Fig.  5,  be  described  on  the  equatorial 
diameter  EQ,  and  let  AF  be  perpendicular  to  the  ellipse  at  A, 
and  BAG  perpendicular  to  EQ.  Put  EQ  =  a,  P p  =  6,  the  true 
latitude  of  A  =  the  angle  EDA  =  L,  and  the  reduced  latitude 
ECA  =  R.  Then, 


CG  tan  R  =  AG  =  DG  tan  L;  hence 

DG  tan  R  ^  , 

— •  = - =-•  But  (Conic  Sections), 

CG  tan  L 


DG  _  b2 
CG  ~  a2 


Therefore 


tan  R  b2 
tan  L  ~  a2’ 


or  a2  :  b2  :  :  tan  L  :  tan  R, 


12.  The  equatorial  and  polar  diameters  of  the  earth, 
and  the  latitude  of  a  place  being  given,  the  radius 
from  the  place  to  the  centre  of  the  earth  may  be  found. 


Put  the  angle  BCG  =  M;  then  to  obtain  the  radius  AC,  we 
have 

BG  =  CG  tan  M,  and  AG  =*  CG  tan  R; 


20 


ASTRONOMY. 


therefore  CGtanM  =  tanM. 

AG  CG  tan  R  tan  R 

but  (Conic  Sections,)  —  — 

V  AG  6 

Hence  — or  tan  M  =  ^  tan  R  =  —  tan  L  = 

tan  R  6  6  b  a2 

b  ,  r 
w  tan  L. 

a 

And  AC  —  s‘n  ^  a  cos  M 

sin  CAG  ~  cos  R 

LONGITUDE. 

13.  We  have  shown  (1)  how  to  obtain  the  latitude 
of  a  place.  But  it  is  evident  that  the  latitude  is  not  of 
itself  sufficient  to  designate  the  situation  of  a  place, 
because  all  the  points  in  a  circle  on  the  earth’s  sur¬ 
face,  parallel  to  the  equator,  have  the  same  latitude. 
Something  more  then  is  necessary  to  designate,  with 
precision,  the  situations  of  places.  As  the  meridians 
cut  the  equator  at  right  angles,  they  are  conveniently 
made  use  of  for  this  purpose. 

14.  A  meridian  which  passes  through  some  par¬ 
ticular  place  is  called  the  First  Meridian .  The  angle 
contained  between  the  first  meridian  and  a  meridian 
through  any  place,  is  called  the  Longitude  of  that 
place.  Longitude  is  measured  by  the  arc  of  the 
equator,  intercepted  between  the  first  meridian  and  the 
meridian  passing  through  the  place,  and  is  called  east 
or  west  according  as  the  latter  meridian  is  to  the  east 
or  west  of  the  first  meridian. 

15.  Different  nations  have  adopted  different  first 
meridians.  The  English  reckon  longitude  from  the 
meridian  which  passes  through  their  Observatory,  at 
Greenwich,  near  London;  and  the  French  from  the 
meridian  of  their  Observatory  at  Paris.  As  there  is 


CHAPTER  IV. 


21 


no  public  Observatory  in  the  United  States,  there  is 
not  a  uniformity  with  respect  to  a  first  meridian. 
Some  reckon  the  longitude  from  the  meridian  of 
Washington,  some  from  that  of  Philadelphia,  and 
others  from  the  meridians  of  other  principal  cities. 
But  for  astronomical  purposes  we  reckon  our  longi¬ 
tude  from  the  meridian  of  Greenwich  or  of  Paris. 

16.  Since  the  diurnal  motion  of  the  stars  is  from 
east  to  west  (1.5),  any  particular  star  must  come  to 
a  given  first  meridian,  sooner  than  to  the  meridian  of  a 
place  which  has  west  longitude,  and  later  than  to  the 
meridian  of  a  place  which  has  east  longitude  (14);  and 
the  difference  of  times  will  be  found  by  allowing  one 
sidereal  hour  for  each  15°  of  longitude,  and  in  the 
same  proportion  for  odd  degrees,  minutes  and  seconds 
(3.5).  It  follows  therefore  that,  if  the  time  at  which 
some  star  passes  the  first  meridian,  be  observed  by  an 
accurate  watch  or  portable  chronometer,  regulated  to 
keep  sidereal  time;  and  if  it  be  then  taken  to  a  place  to 
the  east  or  west  of  the  first  meridian,  and  the  time,  at 
which  the  same  star  passes  the  meridian  of  that  place, 
be  observed  by  it,  the  difference  of  times,  converted 
into  degrees,  by  allowing  15°  to  the  hour,  will  express 
the  longitude  of  the  place. 

There  are  various  other  methods  of  determining  the 
longitudes  of  places,  some  of  which  will  be  noticed  in 
succeeding  parts  of  the  w  ork. 

Table  1.  contains  the  latitudes  of  a  number  of  the 
principal  cities,  and  their  longitudes  from  the  meridian 
of  Greenwich,  expressed  both  in  degrees  and  in  time- 


ASTRONOMY. 


22 

CHAPTER  Y. 

On  Parallax . 

1.  The  directions  in  which  a  body  is  seen  at  the 
same  instant,  from  different  places  on  the  earth’s  sur¬ 
face,  must  in  reality  be  different;  but  the  distances  of 
the  fixed  stars  are  so  immensely  great,  (1.12J,  that  for 
any  one  of  them  the  difference  is  perfectly  insensible. 
This  is  not  the  case  with  the  sun,  moon,  and  planets. 
They  are  sufficiently  near  to  the  earth,  to  have  the  di¬ 
rections  in  which  they  are  seen,  sensibly  influenced  by 
the  situation  of  the  observer.  Astronomers,  therefore, 
in  order  to  render  their  observations  easily  comparable, 
and  for  convenience  in  calculation,  reduce  the  situation 
of  a  body,  as  observed  at  any  place  on  the  earth’s  sur¬ 
face,  to  the  situation  in  which  it  would  appear  from  the 
centre. 

The  observed  situation  of  the  body  is  called  its  Ap¬ 
parent  place,  and  the  situation  in  which  it  would  ap¬ 
pear  from  the  earth’s  centre  is  called  its  True  place. 

2.  The  angle  contained  between  two  right  lines, 
conceived  to  be  drawn  from  a  body,  one  to  the  centre 
of  the  earth  and  the  other  to  the  place  of  the  observer, 
is  called  Parallax .  It  is  also  sometimes  called  Paral¬ 
lax  in  Altitude . 

3.  Let  ADE,  Fig.  6,  represent  the  earth,  considered 
as  a  sphere,  C  its  centre,  A  a  place  on  its  surface,  Z 
the  zenith  of  the  place  A,  and  B  the  situation  of  a 
body;  then  will  ZAB  be  the  apparent  zenith  distance 
of  the  body,  ZCB  its  true  zenith  distance,  and  ABC 
its  parallax  in  altitude. 

The  parallax  ABC  =  ZAB  —  ZCB  =  apparent 
zenith  distance  —  true  zenith  distance  =  90°  —  appa- 


CHAPTER  V.  23 

rent  altitude  —  (90°  —  true  altitude)  =  true  altitude 
—  apparent  altitude. 

4.  The  parallax  in  altitude  of  a  body,  when  its  ap¬ 
parent  zenith  distance  is  90°,  is  called  the  Horizontal 
Parallax . 

5 .  Supposing  a  body  to  continue  at  the  same  dis¬ 
tance  from  the  earth,  the  sine  of  the  parallax  in  alti¬ 
tude  is  equal  to  the  sine  of  the  horizontal  parallax, 
multiplied  by  the  sine  of  the  apparent  zenith  distance. 

Put  R  =  AC  =  mean  radius  of  the  earth, 

D  =  CB  =  distance  of  the  body  from  the  earth’s  centre, 
N  =  ZCB  =  true  zenith  distance, 
p  —  ABC  =  parallax  in  altitude, 
and  w  =  the  horizontal  parallax; 
then  N  +  p  —  ZAB  =  apparent  zenith  distance, 

And  sin  p  =  ^  sin  CAB  ==  ^  sin  ZAB  =  5  sin 
BC  BC  D 

(N  +  p.) 

But  when  N  -f  p  =  20\  p  becomes  w, 

Hence  sin  «■  =  5  sin  90°  = 

D  D 

Consequently  sin  p  =  sin  sin  (N  -f  p.) 

6.  The  distance  of  a  body  from  the  centre  of  the 
earth  is  equal  to  the  radius  of  the  earth,  divided  by  the 
sine  of  the  horizontal  parallax. 

Since  sin 
We  have  D 

Hence,  as  the  radius  of  the  earth  has  been  deter¬ 
mined  (4.9),  when  the  horizontal  parallax  of  a  body  is 


=  5(5), 

R 

sin  w* 


ASTRONOMY. 


■JM* 

known,  its  distance  from  the  centre  of  the  earth  is  easily 
found. 

7.  The  distances  of  the  heavenly  bodies  are  so  great 
that  p,  the  parallax  in  altitude,  and  w,  the  hori¬ 
zontal  parallax,  are  always  very  small  angles;  even 
for  the  moon  which  is  much  the  nearest,  the  value  of 
ar  does  not  at  any  time  exceed  62'.  We  may  there¬ 
fore,  without  sensible  error,  use  p  and  ar  themselves, 
instead  of  their  sines.  If  this  be  done,  the  last  formu¬ 
lae,  in  the  two  preceding  articles,  become, 

p  =  w  sin  (N  4-  p ,) 

and  D  =  —  =  R  — . 

R  w 

8.  In  the  fraction  £  of  the  last  formula,  1  represents 
the  radius  and  ar  the  measure  of  the  horizontal  paral¬ 
lax.  Hence,  in  order  to  render  the  numerator  and  de¬ 
nominator  of  the  fraction  homogeneous,  if  sr  be  ex¬ 
pressed  in  seconds,  we  must  also  express  the  radius  in 
seconds. 

Because  6.2831853  is  the  length  of  the  circumference 
when  the  radius  is  1,  and  1296000  is  the  number  of 
seconds  in  the  circumference;  we  have  6.2S31853  :  i 
::  1296000"  :  206264".8  =  the  length  of  the  radius 
expressed  in  seconds. 

Hence  if  the  value  of  ar  is  expressed  in  seconds, 

D  -  R  206264  -8 

z. r 

9  If  the  meridian  zenith  distances  of  a  body  be  ob¬ 
served  on  the  same  day,  by  two  observers,  remote  from 
each  other  on  the  same  meridian,  and  at  places,  whose 


CHAPTER  V. 


25 

latitudes  are  known,  its  horizontal  parallax  may, 
from  thence  be  determined. 

/ 

Let  AEA'Q,  Fig.  7,  represent  a  meridian  of  the  earth,  con¬ 
sidered  as  a  sphere,  C  its  centre,  EQ  a  diameter  of  the  equator, 
A  and  A'  the  situations  of  two  observers,  Z  and  Z  their  zeniths, 
and  B  the  situation  of  a  body  on  the  meridian. 

Put  L  =  ECZ  =  latitude  of  the  place  A, 

L  =  ECZ'  =  do.  A', 

d  =  ZAB  =  apparent  zenith  distance  at  A, 
and  d'  =  Z'A'B  ==  do.  A'. 

Then, 

'  ACA'  =  ECZ  +  ECZ'  =  L  +  L', 

BAC  =  180°  —  ZAB  =  180°  —  d , 

BA'C  =  180°  —  ZAB  =  180°  —  d', 
and  ABA'  =  360°  —  (ACA'  +  BAC  +  BA'C)  =  d  +  d 
—  (L  -f  L'). 

Again  (7)  ABC  ==  &  sin  d,  A'BC  =  -sr  sin  d', 
and  ABA'  =  ABC  -f  A'BC  =  w  sin  d  -j-  sin  d'. 

Hence  zr  = - - -  =  -  +  -  ~  (L  +  L'l 

sin  d  +  sin  d'  sin  d  -f  sin  d' 

10.  If  the  meridian  zenith  distances  of  a  fixed  star, 
which  passes  the  meridian  nearly  at  the  same  time  with 
the  body,  be  observed,  as  well  as  those  of  the  body, 
the  horizontal  parallax  may  be  obtained,  independent 
t)f  the  latitudes  of  the  places. 

For  if  S  be  the  situation  of  the  star  when  on  the  meridian,  we 
then  know, 

BAS  =  ZAS  —  ZAB  and  BA'S  =  Z'A'B  —  Z'A'S. 

But  ABA'  +  BAS  =  BLS  =  ASA'  +  BA'S, 
or  ABA'  =  BA'S  —  BAS  +  ASA'. 

Or  since  the  angle  ASA'  is  insensible  (1),  we  have, 

ABA'  =  BA'S  —  BAS, 

and  »  =  B_A'S  ~  BAS 
sin  d  -f  sin  d' 


26 


ASTRONOMY. 


11.  It  is  not  necessary  that  the  two  observers  should 
be  on  precisely  the  same  meridian;  for  if  the  meridian 
zenith  distances  of  the  body  be  observed  on  several 
successive  days,  its  change  of  meridian  zenith  distance 
in  a  given  time  will  become  known.  Then  if  the  dif¬ 
ference  of  the  longitudes  of  the  places  is  known  (1.16), 
the  zenith  distance  of  the  body  as  observed  on  one  of 
the  meridians,  may  be  reduced  to  what  it  would  be,  if 
the  observation  had  been  made,  in  the  same  latitude  on 
the  other  meridian. 

In  the  year  1J5 1,  Wargentin,  at  Stockholm,  and  La- 
eaille,  at  the  Cape  of  Good  Hope,  made  the  requisite 
observations  on  the  planet  Mars ,  and  determined  its 
parallax  at  the  time  of  observation  to  be  24". 64. 
Hence,  (8), 


206264".8 
24". 64 


=  H 


x 


8371. 


The  distance  of  Mars  from  the  earth's  centre  was, 
therefore,  at  the  time  of  observation,  8371  times  the 
mean  radius  of  the  earth. 

12.  For  the  moon,  whose  parallax  is  much  greater 
than  that  of  any  other  of  the  heavenly  bodies,  it  is  ne¬ 
cessary  to  take  into  view  the  spheroidical  figure  of  the 
earth. 

Let  the  ellipse  PE  p  2,  Fig.  8,  represent  a  meridian  of  the 
earth,  C  its  centre,  EQ  a  diameter  of  the  equator,  and  Z  and  Z' 
the  true  zeniths  of  the  places  A  and  A'.  The  angle  zAZ  =  CA d 
=  reduction  of  latitude  for  the  place  A,  and  z'AZ'  —  CA 'd'  = 
reduction  oflatitude  for  the  place  A',  may  be  found  (4.11),  and 
thence  the  angle  zAB  =  ZAB  —  zAZ  and  s'A'B  ==  Z'A'B  — 
js'A'Z',  are  known. 

Now  if  w  and  be  the  horizontal  parallaxes  of  the  moon  at 


CHAPTER  V. 


27 


A  and  A',  and  R  and  R'  stand  for  the  radii  CA  and  CA',  we 
have  (7), 

J?  =  D  =  — ;  whence  «•'  ==  «■. 

Let  d  and  d'  stand  for  the  reduced  zenith,  distances  rAB  and 
s'A'B;  then  (7), 

ii' 

ABC  =  -or  sin  d,  A  BC  =  «■'  sin  d'  =  —  sin  d', 


ind  ABA'  =  ABC  +  A  BC  =  sin  d  +  5' 

R 

R  w  sin  d  4.  R'  «■  sin  d'  =  R  x  ABA'. 

R  x  ABA' 


sin  d\ 


Hence 


13.  The  horizontal  parallax  of  the  moon,  to  ah 
observer  at  the  equator,  is  called  the  Equatorial  Pa¬ 
rallax . 


Ifw"  =  the  equatorial  parallax,  and  R"  =  CE  =  the  ra¬ 
dius  of  the  equator,  then, 

,,  R"  R"  x  ABA' 

R  R  sin  d  +  R  sin  d' 

14.  From  observations  made  in  the  year  1751?  by  La- 
caille,  at  the  Cape,  and  Lalande,  at  Berlin,  and  from 
other  methods,  which  have  been  used  for  the  same  pur¬ 
pose,  the  moon’s  equatorial  parallax  is  found  to  Vary 
from  53'  52"  to  6t'  32".  Hence^ 


D  =  R"  =  R"  x  64,  nearly  =  its  greatest  distance, 

and  D  =  R".  =  R"  x  56,  nearly  =*=  its  least  dis¬ 

tance. 

Consequently  the  moon’s  mean  distance  is  about  60 
times  the  equatorial  radius  of  the  earth. 


ASTRONOMY. 


15.  The  mean  equatorial  parallax  of  the  moon  is  £ 
(53'  52"  +  61'  32")  =  57'  42".  But  the  parallax  at 
the  mean  distance  is  only  57'  22". 


Let  D,  D',  and  D",  be  the  least,  greatest,  and  mean  distances 
of  the  moon  from  thfe  earth,  and  «•,  ®-",the  corresponding  pa- 

rallaxes.  Then  (4), 


D  = 


R 


,D 


R' 


and  D"  = 


R' 


sin  w 

i  D  +  D')  =  i  (-j 


sin  w' 
R" 


Sin  vr 


‘  sin  w  sm 


R''  \ 
in  w'/ 


But  D"  =  -J  D  4-  D). 

Hence  -3_  =  1  ( -51  +  -31) 
sin  w"  Vsin  «•  sm  «■'/ 

2  sin  sr  4-  sin  wr'  _  2  sin  ^  4-  73-)  cos  -J  (w — w') 


sm 


sin  sin  w' 


sm  w  sm  ar 


(Ap.  20.) 


smr  = 


sin  «•  sin  «■' 


_ 

sin  i  (ar  4  «•'  COS  (ar  —  -sr') 

As  the  arcs  are  small,  we  may,  without  material  error,  con¬ 
sider  cos  i  (®-  —  «■')  —  1,  and  for  the  other  terms  take  the 
arcs  instead  of  their  sines.  We  shall  then  have, 

*•»  =  _ _  =  2g-g'  fB) 

^  (w  4-  *■')  ar  4 

From  either  of  the  expressions,  A  and  B,  the  value  of  «•"  is 
found  equal  57  22". 


16.  The  sun’s  distance  is  so  great,  that  its  parallax 
can  not  be  determined  with  precision  by  the  preceding 
method.  It  may,  however,  be  shown  to  be  about  8" 
or  9". 

By  a  method  that  will  be  noticed  hereafter,  the  sun’s 
mean  horizontal  parallax  is  found  to  be  8 " .7.  Hence 
its  distance  is, 


Tl  _  20626F.8 

8". 7 


=  23708  x  R 


CHAPTER  VI. 


29 


CHAPTER  VI. 

Apparent  Path  of  the  Sun.— Fixed  Stars. 

1.  The  sun  partakes  with  the  stars  in  the  apparent 
diurnal  motion;  but  the  time  between  his  passing  the 
meridian  on  any  day,  and  his  passing  it  the  next,  is 
found  to  be  greater  than  a  sidereal  day.  The  sun, 
therefore,  appears  to  have  a  motion  eastward  among 
the  fixed  stars.  The  altitude  of  the  sun,  when  on  the 
meridian,  is  not  the  same  on  two  successive  days.  On 
the  20th  of  March  and  22d  of  September,  it  is  about 
the  same  as  the  meridian  altitude  of  the  equator;  from 
the  former  time  to  the  latter,  it  is  greater;  and  during 
the  other  part  of  the  year,  it  is  less.  On  the  2 1st  of 
June  the  sun’s  meridian  altitude  is  greatest,  and  it  then 
exceeds  the  meridian  altitude  of  the  equator  about  23° 
28';  on  the  21st  of  December  it  is  least,  and  is  then 
less  than  that  of  the  equator  by  the  same  quantity  23° 
28'.  The  sun’s  motion  appears,  therefore,  to  be  in  a 
plane,  cutting  the  ecliptic  in  two  opposite  points. 

2.  Let  HZRN,  Fig .  9,  represent  the  meridian^ 
HOR  the  horizon,  P p  the  axis  of  the  heavens,  EQA 
the  equator,  ASFG  the  apparent  path  of  the  sun,  P  the 
north  pole,  and  Z  the  zenith.  Also,  let  S  be  the  situa¬ 
tion  of  some  bright  star,  which,  in  the  latter  part  of 
March  or  in  April,  passes  the  meridian  a  short  time 
before  the  sun.*  Let  the  time  at  which  the  star  passes 
the  meridian  be  observed  by  a  clock,  accurately  regu¬ 
lated  to  sidereal  time.  If  then  the  altitude  SR  of  the 
sun’s  centre,  when  on  the  meridian,  be  observed  and 
corrected  for  refraction  and  parallax,  and  also  the  time 

*  The  brighter  stars  may  be  distinctly  seen  in  the  day  time,  with  an  as¬ 
tronomical  telescope. 


30 


ASTRONOMY. 


be  observed,  we  have  tbe  polar  distance  PS  =  180°  — 
(PH  4-  HS)  =  180°  —  (latitude  of  the  place  4-  alti¬ 
tude  of  the  sun’s  centre),  and  the  angle  BPQ  =  the 
difference  of  times,  converted  into  degrees  (3.6). 

If  similar  observations  be  made  on  the  same  star  and 
the  sun,  a  few  weeks  after,  when  the  sun  has  moved  in 
its  apparent  path  to  S',  we  shall  have  PS'  and  the  an¬ 
gle  EPD.  Consequently  the  angle  BPS'  =  BPD  — 
BPQ,  becomes  known. 

If  the  sun’s  apparent  path  ASF  be  a  great  circle, 
SPS'  will  be  a  spherical  triangle,  in  which  we  know 
the  two  sides  PS,  PS'  and  the  contained  angle  SPS'; 
whence  the  angles  PSS'  and  PS'S  may  be  found. 
Then  in  the  right  angled  triangle  AQS,  we  have  ASQ 
=  PSS'  and  QS  =  90°  —  PS,  with  which  the  angle 
A  may  be  found.  We  may  also  find  the  angle  A  from 
the  triangle  ADS',  in  which  are  given  AS'D  =  180° 
—  PS'S  and  DS'  =  90°  —  PS'.  The  value  of  the 
angle  A,  thus  determined  from  the  two  triangles  AQS 
and  ADS',  is  found  to  be  the  same.  Hence  the  appa¬ 
rent  path  ASG  of  the  sun,  is  a  great  circle.  It  there¬ 
fore  cuts  the  equator  ill  two  points,  A  and  F,  at  the 
distance  of  180°. 

But  little  more  than  half  of  the  circle  is  shown  in  the 
figure,  as  the  whole,  if  accurately  represented,  would 
occupy  too  much  room. 

3.  The  great  circle  which  the  sun  appears  to  de¬ 
scribe,  is  called  the  Eclijptic. 

4.  The  points  in  which  the  ecliptic  cuts  the  equator, 
are  called  the  Equinoctial  Points .  The  time  when  the 
sun  is  at  the  equinoctial  point,  in  his  passage  from  the 
south  to  the  north  side  of  the  equator,  is  called  the 
Vernal  Equinox ;  and  the  time,  when  he  is  at  the  other 
equinoctial  point,  is  called  the  Autumnal  Equinox . 


CHAPTER  VI. 


31 


Tlie  terms  Vernal  Equinox  and  Autumnal  Equinox 
are  frequently  applied  to  the  equinoctial  points  them¬ 
selves. 

5.  The  two  points  in  the  ecliptic,  which  are  at  90c 
distance  from  the  equinoctial  points,  are  called  the  Sol¬ 
stitial  Points.  The  point  T  represents  the  situation  of 
the  solstitial  point  on  the  north  side  of  the  equator;  the 
other  is  on  the  part  of  the  ecliptic  left  out  of  the  figure. 
The  time,  when  the  sun  is  at  the  northern  solstitial 
point  is  called  the  Summer  Solstice ,  and  the  time, 
when  he  is  at  the  southern  solstitial  point,  is  called  the 
Winter  Solstice . 

6.  A  great  circle,  passing  through  the  equinoctial 
points  and  the  poles  of  the  heavens,  is  called  the  Equi¬ 
noctial  Colure .  Another  great  circle,  passing  through 
the  solstitial  points,  is  called  the  Solstitial  Colure . 

7.  The  angle  which  the  ecliptic  makes  with  the 
equator,  is  called  the  Obliquity  of  the  Ecliptic .  The 
obliquity  of  the  ecliptic  is  found  to  be  23°  28'  near¬ 
ly  (S). 

8.  Two  small  circles  parallel  to  the  equator  and 
touching  the  ecliptic  at  the  solstitial  points,  are  called 
the  Tropics .  That,  which  is  on  the  north  side  of  the 
equator  is  called  the  Tropic  of  Cancer ,  and  the  other, 
the  Tropic  of  Capricorn.  Thus  aTb  is  the  tropic  of 
Cancer,  and  ede ,  the  tropic  of  Capricorn. 

9.  Two  small  circles  parallel  to  the  equator  and 
at  a  distance  from  the  poles  equal  to  the  obliquity  of 
the  ecliptic,  are  called  Polar  Circles .  The  one,  about 
the  north  pole,  is  called  the  Arctic  Circle;  the  other, 
about  the  south  pole,  is  called  the  Antarctic  Circle. 
Thus  fgh  is  the  Arctic  Circle,  and  Jcmn ,  the  Antarctic. 

10.  Circles,  corresponding  to  the  tropics  and  polar 
circles,  conceived  to  be  drawn  on  the  earth,  divide  its 


32 


ASTRONOMY. 


surface  into  five  parts,  called  Zones.  The  part  con¬ 
tained  between  the  tropics,  is  called  the  Torrid  Zone , 
the  two  parts  between  the  tropics  and  polar  circles,  are 
called  the  Temperate  Zones ,  and  the  other  two  parts 
within  the  polar  circles,  are  called  the  Frigid  Zones. 

11.  The  ecliptic  is  supposed  to  be  divided  into 
twelve  equal  parts,  which  are  called  Signs.  Each 
sign,  therefore,  contains  30  degrees.  The  division  of 
the  signs  commences  at  the  vernal  equinox,  and  they 
are  numbered  in  the  direction  of  the  sun’s  apparent 
motion  in  the  ecliptic.  The  signs  of  the  ecliptic  are, 
sometimes,  designated  by  names  or  characters,  instead 
of  numbers. 

The  names  of  the  twelve  signs  with  their  correspond¬ 
ing  numbers,  and  the  characters  by  which  they  are  usu¬ 
ally  denoted,  are, 


s. 

0. 

Aries 

r. 

s. 

6.  Libra 

1. 

Taurus 

8. 

7.  Scorpio 

*1 

2. 

Gemini 

n. 

8.  Sagittarius 

t 

3. 

Cancer 

23. 

9.  Capricornus 

V? 

4. 

Leo 

si. 

10.  Aquarius 

AW' 

(VW 

5. 

Virgo 

11.  Pisces 

X 

Aries,  Taurus,  Gemini,  Cancer,  Leo,  and  Virgo  lie 
on  the  north  side  of  the  equator  and  are  called  North¬ 
ern  Signs.  The  others  lie  on  the  south  side,  and  are 
called  Southern  Signs. 

Capricoruus,  Aquarius,  Pisces,  Aries,  Taurus  and 
Gemini  are  called  Ascending  Signs ,  because  w  hile  the 
sun  is  in  them,  his  meridian  altitude  continually  in¬ 
creases.  Cancer,  Leo,  Virgo,  Libra,  Scorpio  and 
Sagittarius  are  called  Descending  Signs ,  because  the 


CHAPTER  VI.  33 

sun’s  meridian  altitude  continually  decreases,  while  he 
is  in  them. 

IS.  A  zone  of  the  heavens  extending  in  breadth  to 
8  or  9°  on  each  side  of  the  ecliptic,  is  called  the  Zo¬ 
diac.  Within  the  zodiac,  all  the  planets  perform  their 
motions,  except  three  of  those  recently  discovered. 

13.  Any  great  circle,  which  passes  through  the 
poles  of  the  ecliptic,  is  called  a  Circle  of  Latitude. 

14k  The  arc  of  the  ecliptic,  intercepted  in  the  order 
of  the  signs,  between  the  vernal  equinox  and  a  circle  of 
latitude,  which  passes  through  a  star,  is  called  the 
Longitude  of  the  star.  And  the  arc  of  the  circle  of 
latitude,  intercepted  between  the  star  and  the  ecliptic, 
is  called  the  Latitude  of  the  star.  Latitude  is  said  to 
be  north  or  south,  according  as  the  body  is  on  the  north 
or  south  side  of  the  ecliptic. 

15.  Any  great  circle,  which  passes  through  the 
poles  of  the  equator,  is  called  a  Circle  of  Declination. 

16.  The  arc  of  the  equator,  intercepted  between  the 
vernal  equinox  and  a  declination  circle,  which  passes 
through  a  star,  is  called  the  1 Right  Ascension  of  the 
star.  And  the  arc  of  the  circle  of  declination,  inter¬ 
cepted  between  the  star  and  the  equator,  is  called  the 
Declination  of  the  star.  Declination  is  said  to  be  north 
or  south,  according  as  the  body  is  on  the  north  or  south 
side  of  the  equator. 

Longitude  and  Right  Ascension  are  both  reckon¬ 
ed  from  the  vernal  equinox,  round  to  it  again,  in  the 
order  of  the  signs. 

17.  The  situations  of  the  fixed  stars,  are  generally 
expressed  by  right  ascension  and  declination,  and  those 
of  the  sun,  moon  and  planets,  by  longitude  and  latitude. 
W  ith  the  obliquity  of  the  ecliptic  known,  the  longitude 
and  latitude  of  a  body  may  be  obtained  from  the  right 

6 


ASTRONOMY. 


34 

ascension  and  declination,  by  means  of  spherical  Tri¬ 
gonometry.  On  the  contrary  from  tlie  longitude  and 
latitude,  the  right  ascension  and  declination  may  be 
found. 


18.  Let  EQ,  Fig.  10,  represent  the  equator,  EC  the  ecliptic, 
P  and  P'  their  poles,  E  the  vernal  equinox,  PSR  a  circle  of  de¬ 
clination  and  PS  L  a  circle  of  latitude,  both  passing  through  a 
body  at  S.  Then  will  ER  be  the  right  ascension,  RS  the  decli¬ 
nation,  EL  the  longitude  and  LS  the  latitude  of  the  body. 


Put  R  =  ER  =  right  ascension, 

D  =  RS  =  declination, 

L  =  EL  =  longitude, 
a  —  LS  =  latitude, 
of  =  REL  =  obliquity  of  the  ecliptic, 
x  =  RES, 
and  y  =  LES. 


19.  When  the  right  ascension  and  declination  are  given  to  find 
the  longitude  and  latitude ,  we  have, 

tang  RES  =  tang  ES  = 

sin  ER  cos  RES 

tang  EL  =  cos  LES  tang  ES  =  cos  LES**nf ER; 

cos  RES 

and  tang  LS  =  tang  LES  sin  EL. 

Or  tang  *  =  tang  L  = 

sin  R  cos  x 


and  tang  a  =  tang  (a; — *>)  sin  L. 


If  attention  be  given  to  the  rules  for  trigonometrical  signs,  and  tang 
D  be  considered  negative  when  the  declination  is  south ,  these  for¬ 
mulae  will  apply,  whatever  be  the  situation  of  the  body;  observing 
that  the  longitude  and  right  ascension  are  always,  either,  both  be¬ 
tween  90°  and  270°  or  both  between  270"  and  90%  and  that,  when 
the  tang  a  comes  out  negative ,  the  latitude  is  south. 

Let  S'  be  the  sun’s  place  in  the  ecliptic;  then  ES'  =  sun’s 
longitude.  Hence 


CHAPTER  VI. 


35 


.  .  ,  x  Tio/  tan  ER 

tan.  Sun’s  longitude  =  tan  Eb  =  — — 

tan  Sun's  right  ascension 

cos  Obliquity  of  the  ecliptic 

20.  When  the  longitude  and  latitude  are  given  to  find  the  right 
ascension  and  declination ,  we  have , 

™  C0S  EES  tan§  EE- 

tang  ER  =  cos  RES  tang  ES  = - —  > 

6  &  cos  LES 

and  tang  RS  =  tang  RES  sin  ER. 

/a  x  tang  a  .  0  cos  («.+  •}  tang  L. 

Or  tang  y  =  — ;  tang  R  =  — 

sin  L  b  cos  y 

and  tang  D  =  tang  (y+*>)  sin  R. 


When  the  latitude  is  south,  tang  a  must  be  considered  nega¬ 
tive;  and  if  tang  D  come  out  negative,  the  declination  will  be 
south. 

For  the  sun  we  have,  tan  Sun's  right  ascension  ==  tan  ER  = 
cos  RES'  tan  ES'  =  cos  Obliquity  of  the  ecliptic  x  tan  Sun's 
longitude;  and  sin  Sun's  decl.  =  sin  RS'  =  sin  RES'  sin  ES' 
=  sin  Obliquity  of  the  ecliptic  x  sin  Sun's  long . 


SI.  The  angle  contained  between  a  circle  of  latitude 
and  circle  of  declination,  both  passing  through  the  sun 
or  a  star,  is  called  the  tingle  of  Position  of  the  star. 

If  P'P  be  produced  to  meet  the  equator  in  N,  then  in  the  tri¬ 
angle  P'SP,  P'S  =  complement  of  latitude,  PS  =  complement 
of  declination,  P'P  =  obliquity  of  the  ecliptic,  P'PS  ==  180  — 
NPR  =  180  —  NR  =  180’  —  (EN  —  ER)  =  180’  —  (90* 
—  ER)  =  90°  +  ER  =  90"  +  right  ascension,  and  PP'S  = 
EM  —  EL  =  90°  —  longitude.  With  any  three  of  these  five 
parts  given,  the  angle  of  Position  PSP'  may  be  found. 

When  the  longitude,  latitude,  and  obliquity  of  the  ecliptic  are 
given,  we  have,  putting  S  =  PSP'  the  angle  of  position,  ( App.  37). 


36 


ASTRONOMY. 


COt  S  = 


cot  PP'  sin  P'S  —  cos  P  S  cos  PP'S 


sin  PP  S 

cot  a  cos  a  —  sin  a  cos  (90  —  L) 
sin  (90  —  L) 
cot  a  cos  a  —  sin  a  sin  L 


sin  L 


(c- 

\si 


cos  L 
cot  a 


COS  A 


sin  a 


,  T  /cot 

=  tan  L.  [  - — cos  A 


cos  L  \sin  L 

-  —  sinAV 

\sin  L  / 

Make  tan  z  =  sin  L  tan  a. 

_ 1  1 

sin  L 


Then  C0~  = 


sin  L  tan 


—  cot  z  — 
tan  z  sin 


cos  z 


Hence  cot  S  =  tan  L.  fcos  z.  cos  a  —  sin  a^ 
Vsin  z  } 


tan  L  /  . 

— - .  (cos  z  cos  a  —  sin  z  sm  a). 

sin  2 


But  App.  14)  cos  z  cos  a  —  sin  z  sin  a  —  cos  ( z  -}-  a). 

(A) 


Therefore  cot  S  =  .  cos  (z  +  a). 

sin  z 


When  the  longitude,  declination,  and  obliquity  of  the  ecliptic, 
are  given,  we  have 

sin  PP'  sin  PP'S  cos  L  sin 


sin  PS  cos  D 

For  the  sun,  a  =  o,  and  the  formula 


(B) 


cot  S  = 


cot  S  = 


cot  eo  cos  a  —  sin  a  sin  L 


cos  L 


,  becomes, 


cot  # 
cos  L’ 


Or  tan  S  =  cos  ~  _  £  tan  a 
cot  u 


(C) 


It  is  easy  to  see  that  the  northern  part  P'S  of  the  circle  of  lati¬ 
tude  is  to  the  west  of  the  northern  part  PS  of  the  circle  of  decli¬ 
nation,  when  the  longitude  is  less  than  90  or  more  than  270°,  and 
to  the  east  when  it  is  between  90°  and  270°. 


CHAPTER  VI. 


37 


SITUATIONS  OF  THE  FIXED  STARS. 

22.  Iu  order  to  distinguish  the  fixed  stars  from  each 
other,  the  ancients  supposed  the  figures  of  men,  ani¬ 
mals  or  other  objects  to  be  drawn  on  the  concave  sur¬ 
face  of  the  heavens.  This  mode  of  distinction  is  still 
used.  The  group  of  stars  contained  within  the  con¬ 
tour  of  any  such  figure,  is  called  a  Constellation . 

The  following  tables  exhibit  the  names  of  the  prin¬ 
cipal  constellations. 

1.  ANCIENT  CONSTELLATIONS. 


Northern . 


1.  Ursa  Minor, 

The  Little  Bear. 

2.  Ursa  Major, 

The  Great  Bear. 

3.  Draco, 

The  Dragon. 

4.  Caepheus, 

Caepheus. 

5 .  Bootes, 

Bootes. 

6.  Corono  Borealis, 

The  Northern  Crown. 

7*  Hercules, 

Hercules  kneeling. 

8.  Lyra, 

The  Lyre. 

9.  Cygnus, 

The  SwTan. 

10.  Cassiopea, 

The  Lady  in  her  Chair. 

11.  Perseus, 

Perseus. 

12.  Auriga, 

The  Wagoner. 

13.  Serpentarius, 

Serpentarius. 

14.  Serpens, 

The  Serpent. 

15.  Sagitta, 

The  Arrow. 

16.  Aquila, 

The  Eagle. 

17-  Delphinus, 

The  Dolphin. 

18.  Equulus, 

The  Horse’s  Head. 

19.  Pegasus, 

The  Flying  Horse. 

20.  Andromeda, 

Andromeda. 

21.  Triangulum, 

The  Triangle. 

38 


ASTRONOMY, 


Constellations  of  the  Zodiac . 


22.  Aries, 

The  Ram. 

23.  Taurus, 

The  Bull. 

24).  Gemini, 

The  Twins. 

25.  Cancer, 

The  Crab. 

26.  Leo, 

The  Lion. 

27.  Virgo, 

The  Virgin. 

28.  Libra, 

The  Scales. 

29.  Scorpio, 

The  Scorpion. 

30.  Sagittarius, 

The  Archer. 

31.  Capricornus, 

The  Goat. 

32.  Aquarius, 

The  Water-bearer. 

33.  Pisces, 

The  Fishes. 

Southern . 

34.  Cetus, 

The  Whale. 

35.  Orion, 

Orion. 

36.  Eridanus, 

Eridanus. 

37.  Lepus, 

The  Hare. 

38.  Canis  Major, 

The  Great  Dog. 

39.  Canis  Minor, 

The  Little  Dog. 

40.  Argo, 

The  Ship. 

41.  Hydra, 

Tlie  Hydra. 

42.  Crater, 

The  Cup. 

43.  Corvus, 

The  Crow. 

44.  Centaurus, 

The  Centaur. 

45.  Lupus, 

The  Wolf. 

46.  Ara, 

The  Altar. 

47.  Corona  Australis. 

,  The  Southern  Crown. 

48.  Piscis  Australis, 

The  Southern  Fish. 

2.  NEW  SOUTHERN  CONSTELLATIONS. 

4.  Columba  Noachi,  Noah’s  Dove. 


CHAPTER  VI. 


39 


2.  Robur  Carolipum, 

3.  Gras, 

4.  Phoenix, 

5.  Indus, 

6.  Pavo, 

7.  Apus,  Ms  Indicay 

8.  Apis,  Musca, 

9.  Chamelion, 

10.  Triangulum  Australe, 

11.  Piscis  volans,  Passer , 

12.  Dorado,  Xiphias , 

13.  Toucan, 

14.  Hydrus, 


The  Royal  Oak. 

The  Crane. 

The  Phoenix. 

The  Indian. 

The  Peacock. 

The  Bird  of  Paradise 
The  Bee  or  Fly. 

The  Chamelion. 

The  Southern  Triangle. 
The  Flying  Fish. 

The  Sword  Fish. 

The  American  Goose. 
The  Water  Snake. 


3.  HEVELIUS’  CONSTELLATIONS. 


Made  out  of  the  Unformed  Stars . 

The  Lynx. 


1.  Lynx, 

2.  Leo  Minor, 

3.  Coma  Berenices, 

4.  Asteron  and  Chara, 

5.  Cerberus, 

6.  Yulpecula  and  Anser, 

7.  Antinous, 

8.  Scutum  Sobieski, 

9.  Lacerta, 

10.  Camelopardalis, 

11.  Monoceros, 

12.  Sextans, 


The  Little  Lion. 
Berenice’s  Hair. 
The  Greyhounds. 
Cerberus. 

The  Fox  and  Goose, 
Antinous. 

Sobieski’s  Shield. 
The  Lizard. 

The  Camelopard. 
The  Unicorn. 

The  Sextant. 


23.  The  stars  of  a  constellation  are  distinguished 
by  the  letters  of  the  Greek  alphabet,  which  are  ap¬ 
plied  to  them  according  to  their  apparent,  relative  size* 
The  principal  star  in  the  constellation  is  named  *,  the 


40 


ASTRONOMY. 


second  in  order  the  third  y,  and  so  on.  When  the 
number  of  stars  in  a  constellation,  exceeds  the  letters 
in  the  Greek  alphabet,  the  letters  of  the  Roman  alpha¬ 
bet  are  applied  to  the  remainder  in  the  same  manner; 
and  when  these  are  not  sufficient,  the  numbers  1,  2,  3, 
&c.  are  used  to  designate  those  that  are  left. 

Some  of  the  fixed  stars  have  particular  names,  as 
Sirius,  Aldebaran,  Arcturus,  Regulus,  &c. 

24.  The  stars  are  also  divided  into  classes,  depend¬ 
ing  on  their  apparent  magnitudes.  Those  of  the  first 
class,  are  called  stars  of  the  first  magnitude ,  those  of 
the  second,  stars  of  the  second  magnitude ,  and  so  on, 
to  stars  of  the  sixth  magnitude ,  which  includes  all 
those  that  are  just  visible  to  the  naked  eye.  Those 
stars  which  are  not  visible  without  the  aid  of  a  tele¬ 
scope,  are  called  telescopic  stars. 

25.  In  the  triangle  ASQ,  Fig.  9.  having  QS  and 
the  angle  ASQ  (2),  AQ  may  be  found;  then  AQ  — 
QB  =  AB  =  the  right  ascension  of  the  star  s.  When 
the  right  ascension  of  one  star  is  obtained,  the  right 
ascension  of  any  other  may  be  found  by  observing  the 
difference  between  the  time  of  its  passing  the  meridian 
and  the  time  of  the  known  star  doing  the  same. 

Let  s'  be  the  situation  of  the  star  s  when  on  the  me¬ 
ridian,  then  its  declination  Bs  =  Qs'  =  Rs'  —  RQ  = 
Rs'  —  (90°  —  PH)  =  R s'  +  PH  —  90°  =  the  cor¬ 
rect  meridian  altitude  of  the  star  +  latitude  of  the 
place  —  90°. 

When  the  right  ascensions  and  declinations  of  the 
stars  have  been  obtained,  their  longitudes  and  lati¬ 
tudes  may  be  calculated  (19) 

26.  A  table  containing  a  list  of  fixed  stars,  desig¬ 
nated  by  their  proper  cliaracters,  and  giving  their  right 


CHAPTER  VI.  41 

ascensions  and  declinations  or  tlieir  longitudes  and 
latitudes,  is  called  a  Catalogue  of  those  stars. 

27.  Hipparchus  began  the  first  catalogue  of  the  fixed 
stars  120  years  before  the  Christian  era.  This  cata¬ 
logue,  with  some  additions,  was  afterwards  published 
by  Ptolemy,  and  contained  the  situations  of  1022 
stars.  The  Britannic  Catalogue,  published  by  Flam- 
stead,  in  1689,  contained  the  situations  of  3000  stars. 
Since  that  period  various  other  catalogues  have  been 
published,  some  of  which  are  very  extensive.  Bode’s 
Atlas  and  Catalogue  contain  the  situations  of  17^000 
stars.  The  Catalogues  ofLacaille,  Bradley,  Mayer, 
and  Maskelyne  are  not  extensive,  but  they  are  valued 
for  their  accuracy  . 

All  the  fixed  stars,  visible  to  the  naked  eye,  with 
some  others,  are  represented  on  celestial  globes  of  12 
or  18  inches  diameter. 

28.  The  number  of  stars  visible  with  the  best  tele¬ 
scopes,  amounts  to  several  millions:  but  the  number  vi¬ 
sible  to  the  naked  eye,  is  much  less  than  is  generally 
supposed  by  those  who  only  judge  from  the  impressions 
made,  when  noticing  them  on  a  fine  evening.  The 
number  thus  visible  at  any  one  time  above  the  horizon 
does  not  much  exceed  1000. 

29.  Many  of  the  stars,  which  to  the  naked  eye,  or 
through  telescopes  of  small  power,  appear  single,  are 
found  with  high  magnifiers  to  consist  of  two,  three,  or 
more  stars,  extremely  near  to  each  other. 

30.  The  fixed  stars  are  not  entirely  exempt  from 
change.  Several  stars,  which  are  mentioned  bv  the  an¬ 
cient  astronomers,  have  uow  ceased  to  be  visible,  and 
some  are  now  visible  to  the  naked  eye,  which  are  not 
in  the  ancient  catalogues. 

31.  Many  spaces  are  discovered  in  the  heavens, 

ry 

4 


ASTRONOMY. 


which  are  faintly  luminous,  and  shine  with  a  pale 
white  light.  On  applying  to  them  telescopes  of  great 
power,  they  are  found  to  consist  of  a  multitude  of  small 
stars,  distinctly  separate,  but  very  near  to  each  other. 
These  are  called  Nebidce.  The  Milky  Way  is  a  space 
of  this  kind,  and  is  visible  to  the  naked  eye. 

PRECESSION  OF  THE  EQUINOXES. 

32.  By  comparing  catalogues  of  the  same  fixed  stars, 
formed  at  different  periods,  it  is  found  that  their  lati¬ 
tudes  continue  very  nearly  the  same,  but  that  all  their 
longitudes  increase  at  the  rate  of  50".  1  in  a  year.  As 
the  latitude  of  a  star  is  its  distance  from  the  ecliptic 
(14),  it  follows  from  the  first  mentioned  circumstance, 
that  the  plane  of  the  ecliptic  remains  fixed,  or  very 
nearly  so,  with  respect  to  the  situations  of  the  fixed 
stars. 

33.  The  longitude  of  a  star  being  the  arc  of  the 
ecliptic,  intercepted  in  the  order  of  the  signs,  between 
the  vernal  equinox,  and  a  circle  of  latitude  passing 
through  the  star  (14),  it  follows  from  the  circumstance 
of  all  the  stars  having  the  same  increase  of  longitude 
(32),  that  the  vernal  equinox  must  have  a  motion  along 
the  ecliptic  in  a  direction  contrary  to  the  order  of  the 
signs,  amounting  to  50".l  in  a  year.  As  the  autumnal 
equinox  is  always  at  the  distance  of  180°  from  the  ver¬ 
nal  equinox  (2  and  4),  it  must  have  the  same  motion. 
This  retrograde  motion  of  the  equinoctial  points,  is 
called  the  Precession  of  the  Equinoxes . 

34.  As  the  ecliptic  remains  fixed  (32),  its  pole  must 
also  continue  in  the  same  place;  but  the  equator  and  its 
pole  must  change  their  situations,  otherwise  there  could 
not  be  a  motion  in  the  equinoctial  points.  Let  E'C', 
Fig .  11,  be  the  ecliptic,  P  its  pole,  jp'pT)  a  circle  about 


CHAPTER  YI. 


43 

the  pole  P,  at  a  distance  equal  to  tlie  obliquity  of  the 
ecliptic^  EQG  the  equator,  and  P/;LQ  the  solstitial 
colure.  Then  because  E  is  the  pole  of  PLQ,  the  pole 
of  the  equator  EQC  is  in  PLQ;  it  is  also  in  the  small 
circle  p'pT);  it  is  therefore  at  p.  Now,  if  the  vernal 
equinox  E,  move  by  a  retrograde  motion  to  E',  the 
solstitial  point  L  will  move  a  like  distance  to  L';  there¬ 
fore  E'Q'C'  will  then  be  the  equator,  Pp'L'Q'  the  sol¬ 
stitial  colure,  and  p'  the  pole  of  the  equator.  Hence 
the  pole  of  the  equator  has  a  retrograde  motion,  in  a 
small  circle  about  the  pole  of  the  ecliptic,  and  at  a  dis¬ 
tance  from  the  latter  pole,  equal  to  the  obliquity  of  the 
ecliptic. 

The  precession  of  the  equinoxes  being  only  j50".1  in 
a  year,  it  must  require  25869  years  for  them  to  move 
through  the  whole  of  the  ecliptic;  and  the  pole  of  the 
equator  will  evidently  require  the  same  time  to  make 
its  retrograde  revolution  about  the  pole  of  the  ecliptic. 

35.  The  change  in  the  situation  of  the  equator,  which 
produces  the  precession  of  the  equinoxes,  must  also 
produce  changes  in  the  right  ascensions  and  declina¬ 
tions  of  the  stars.  These  changes  are  different,  accord¬ 
ing  to  the  situations  of  the  stars  with  respect  to  the 
equator  and  equinoctial  points.  The  change  which 
takes  place  in  the  right  ascension  or  declination  of  a 
star  in  the  course  of  a  year,  is  called  the  Annual  Varia¬ 
tion  in  right  ascension  or  declination. 

Let  s  be  the  situation  of  a  star,  psab  a  declination 
circle,  when  EQC  is  the  situation  of  the  equator,  EE' 
the  annual  precession  in  longitude  =  50".l  (32),  and 
E m  and  p'sa'  declination  circles,  when  the  situation  of 
the  equator  is  E'Q'C'.  Then  the  difference  between  a's 
and  as ,  is  the  annual  variation  in  declination  of  the  star 
a,  and  the  difference  between  E'a'  and  E a,  is  its  an- 


44 


ASTRONOMY. 


nual  variation  in  right  ascension.  The  annual  varia¬ 
tions  in  right  ascension  and  declination,  may  he  ex¬ 
pressed  in  formulae,  involving  only  the  right  ascension 
and  declination  of  the  star,  and  the  obliquity  of  the 
ecliptic. 


ANNUAL  VARIATION  IN  DECLINATION. 


36.  Let  sn  be  equal  to  sp'  and  np'  be  joined  by  the  arc  of  a 
great  circle;  then  as  the  arcs  pp'  and  p'n  are  evidently  very  small, 
we  may,  without  sensible  error,  consider  pp '  as  the  arc  of  a  great 
circle,  and  the  angle  pnp'  as  a  right  angle,  excepting,  with  respect 
to  the  latter,  the  case  in  which  the  star  is  very  near  the  pole. 
We  may  also  consider  any  very  small  arc  as  equal  to  its  sine  or 
tangent. 

Put  ea  —  pP  =  E'Ehi  =  obliquity  of  the  ecliptic, 

R  =  Ea  =  right  ascension, 
and  v  =  variation  in  declination.  Then 

,  pYp'  sin  wP  50".  1  sin  ,  . 

sin  pp' P  sin  90° 

pn  —  pp'  cos  p'ps  =  50".  1  sin  a  cos  p'ps. 

But  p'ps  =  90°  —  apQ  =  EQ  —  aQ  =  Ea-  =  R, 
and  pi  =  sp  —  sn  =  sp  —  sp*  =  903  —  as  —  (90  —  a's) 
=.  a's  —  as  =  v. 

Therefore  v  =  50".  1  sin  cos  R. 

When  the  declination  is  north,  as  in  the  figure,  the  sign  of  v  is 
the  same  as  the  sign  of  cos  R;  but  when  the  declination  is  south, 
the  sign  of  v  must  be  contrary  to  that  of  cos  R. 


ANNUAL  VARIATION  IN  RIGHT  ASCENSION. 


37.  Let  D  =  as  =  declination  of  the  star,  and  V  =  its  an¬ 


nual  variation  in  right  ascension.  Then 


V  =  E 'a'  — Ea  =  E'a'  —  mb  =  E  m  -p  a'b.; 
E 'm  =.  E'E  cos  EE'm  --=  50"  1  cos 
p'n  =  pp'  sin  p'pn  =  50".  1  sin  ^  sin  R; 
p'n  50".  1  sin  a  sin  R 


psp'  = 


sin  p'i 


■  ;  and 


cos  a 


CHAPTER  VI. 


45 


a’b  =  a'sb  gin  a’s  =  psp'  sin  a's  = 


50". 1  sin  a  sin  R  sin  a' 
cos  a's 


=  50".  1  sin  <ysinRtan  a's. 

As  the  quantity  50".  1  sin  u  sin  R,  which  is  multiplied  by  tan 
a's,  is  very  small,  and  as  the  difference  between  a's  and  as,  can 
only  be  a  few  seconds,  we  may,  without  sensible  error,  consider 
tan  a's  =  tan  as  —  tan  D.  Therefore 
a'b  =  50". 1  sin  a  sin  Rtan  D,  and 

V  =  E 'm  -f  a'b  =  50".  1  cos  &>  -f  50".  1  sin  «  sin  R  tan  D. 

When  the  declination  is  south,  tan  D  must  be  taken  negative. 
The  second  term  of  the  value  of  V  is  negative  when  the  right  as¬ 
cension  is  less  than  180  and  declination  south,  or  when  the  right 
ascension  is  more  than  180°  and  declination  north.  In  either  of 
these  cases,  when  sin  R  tan  D  is  so  great  as  to  make  the  second 
term  exceed  the  first,  the  value  of  V  is  negative. 

E'm  =  50".  1  cos  a  =  46",  is  the  annual,  retrograde  motion  of 
the  equinoctial  points  along  the  equator. 


38.  In  catalogues  of  the  fixed  stars,  which  express 
their  situations  by  right  ascensions  and  declinations, 
the  annual  variations  in  these  are  also  stated,  with 
their  proper  signs.  In  some  catalogues  the  north  po¬ 
lar  distances  of  the  stars  are  given,  instead  of  the  de¬ 
clinations.  The  variations  will  be  the  same,  except 
that  the  sign  will  be  different  when  the  north  polar  dis¬ 
tance  is  less  than  90°. 

39.  With  the  right  ascension  and  declination  of  a 
star  for  a  given  time  and  their  annual  variations,  its 
right  ascension  and  declination  may  be  found,  with 
considerable  accuracy,  for  a  time  a  few  years  later  or 
earlier.  Put  t  =  the  number  of  years,  then  t.  V  = 
its  change  in  right  ascension  and  t.  v  =  its  change  in 
declination,  nearly.  If  the  time  for  which  the  right 
ascension  and  declination  are  required,  is  after  the 
given  time,  t.  V  and  t.  v  must  be  applied,  with  their 
signs  as  determined  by  the  preceding  formulse;  but,  if 


46 


ASTRONOMY . 


it  is  before  the  given  time,  t  must  be  considered  nega¬ 
tive,  which  will  change  the  signs  of  t.  V  and  t .  v. 

When,  from  the  right  ascension  and  declination  of  a 
star  for  a  given  time,  its  right  ascension  and  declina¬ 
tion  are  required  with  accuracy,  for  a  time  several  years 
earlier  or  later,  they  can  be  found  by  rigorous  formu¬ 
lae,  which  have  been  investigated  for  the  purpose;  or 
with  nearly  the  same  facility,  by  calculating  the  lon¬ 
gitude  and  latitude  of  the  star  for  the  given  time  (19), 
adding  to  the  longitude  the  precession  in  longitude, 
which  will  be  the  product  of  50".  1  by  the  interval  of 
time  expressed  in  years  and  parts,  and  then  with  the 
longitude  thus  obtained  and  the  latitude,  calculating 
the  right  ascension  and  declination. 

40.  In  consequence  of  the  precession  of  the  equi¬ 
noxes,  the  twelve  signs  of  the  ecliptic,  which  about 
S000  years  ago,  respectively  corresponded  with  the 
twelve  constellations  of  the  zodiac,  bearing  the  same 
names,  have  receded  so  far  that  the  sign  Taurus, 
now  corresponds  nearly  with  the  constellation  Aries. 

41.  In  the  preceding  investigations,  we  have  con¬ 
sidered  the  plane  of  the  ecliptic  as  fixed  and  the  obli¬ 
quity  of  the  ecliptic  as  continuing  always  the  same. 
This,  though  very  nearly,  is  not  strictly  true.  A  com¬ 
parison  of  accurate  observations,  made  at  long  intervals 
of  time,  proves  that  each  is  subject  to  a  slight  change. 
These  changes  will  be  noticed  in  a  succeeding  chapter. 

CHAPTER  VII. 

Sun’s  Apparent  Orbit — Kepler’s  Laws — Kepler’s 
Problem . 

1.  It  has  been  shown  (6.37)  that  the  vernal  equinox 
has  a  retrograde  motion  along  the  equator  of  46"  a 


CHAPTER  VII. 


47 


year.  This  is  its  mean  motion.  It  has  been  found 
from  accurate  observations  that  this  motion  is  not  uni¬ 
form. 

The  place  at  which  the  vernal  equinox  would,  at  any 
time,  be,  if  its  motion  was  uniform,  is  called  its  Mean 
place,  or  the  Mean  Equinox, 

2.  The  motion  of  the  mean  equinox  along  the  equa¬ 

tor,  being  46"  a  year,  its  motion  in  one  day  must  be 
|  of  a  second  nearly,  which  corresponds  to  T|o  of  a 
second  in  time  (3.6).  If  therefore  on  any  particular 
day  the  mean  equinox  be  on  the  meridian  at  precisely 
the  same  instant  with  some  fixed  star,  it  would,  in  con¬ 
sequence  of  its  retrograde  motion,  come  to  the  meridian 
on  the  succeeding  day  of  a  second  earlier  than  the 
star.  ^ 

3.  Some  astronomers,  with  a  view  to  convenience 
in  observing  the  right  ascensions  of  the  heavenly  bo¬ 
dies,  regulate  their  clocks  so  as  to  mark  0  h.  0  m 
0  sec.  when  the  mean  equinox  is  on  the  meridian,  and 
they  call  the  interval  between  two  of  its  consecutive 
passages  over  the  meridian,  a  Sidereal  Day. 

The  term  sidereal  day  is  now  generally  used  as 
here  defined,  and  is  to  be  thus  understood  in  the  fol¬ 
lowing  parts  of  the  work.  But  on  account  of  the  very 
small  difference  between  its  length  and  the  length  of 
the  sidereal  day  as  defined  in  a  preceding  chapter 
(3.5),  we  may  consider  them  as  equal  in  all  cases  that 
regard  observations  made  during  a  single  day  or  a 
small  number  of  days. 

4.  The  time  between  two  consecutive  passages  of 
the  sun’s  centre,  over  the  meridian,  is  called  a  True 
Solar  Day .  In  consequence  of  the  sun’s  motion  east¬ 
ward  among  the  fixed  stars  (6.1),  the  length  of  a  solar 
day  is  greater  than  that  of  a  sidereal  day. 


48 


ASTRONOMY. 


It  is  ascertained  by  observations  that  the  length  of 
a  solar  day  is  different  at  different  times  in  the  year, 
imt  that  at  the  same  time  in  different  years  it  is  very 
nearly  the  same.  By  comparing  the  number  of  solar 
days  that  elapse  from  the  time  that  the  sun  passes  the 
meridian  on  a  given  day  in  any  year,  to  the  time  of  its 
passage  on  the  same  day  in  some  succeeding  year, 
with  the  number  of  sidereal  days  and  parts  of  a  day, 
that  elapse  during  the  same  time,  it  is  found  that  the 
mean  length  of  a  solar  day,  called  a  Mean  Solar 
Day,  is  equal  to  24  h.  3  m.  5 6. 555  sec.  of  sidereal 
time. 

5 .  The  ratio  of  24  h  :  24  h.  3  m.  56.555  sec.  is  the 
same  as  1  :  1.0027379,  which  is  therefore  the  ratio  of 
a  mean  solar  day,  to  a  sidereal  day.  Hence  to  reduce 
a  given  portion  of  mean  solar  time  to  the  corresponding 
sidereal  time,  we  must  multiply  by  1.0027379;  aud  on 
the  contrary  to  reduce  sidereal,  to  mean  solar  time,  we 
must  divide  by  the  same  number. 

The  excess  of  a  mean  solar  day  above  a  sidereal 
day  is  3  m.  56.555  sec.  in  sidereal  time;  and  in  mean 
solar  time  it  is  3  m.  55.91  sec. 

6.  By  observing  the  meridian  altitude  of  the  sun’s 
centre,  and  correcting  it  for  refraction  and  parallax, 
its  north  polar  distance  may  be  determined  (6.2).  If 
this  be  done  on  several  successive  days,  about  the 
20th  of  March,  it  will  be  found,  either  that  on  some 
one  of  these  days,  the  north  polar  distance  is  exactly 
90°,  and  consequently  that  the  sun  is  then  at  the 
equinox,  or  which  is  much  more  probable,  that  on  the 
first  of  some  two  consecutive  days  the  north  polar  dis¬ 
tance  is  greater,  and  on  the  second  less,  than  90°. 
From  these  observations,  the  time  that  the  sun  is  at 
the  vernal  equinox  may  be  determined. 


CHAPTER  VII. 


49 


Let  A,  Fig.  12,  be  the  sun’s  place  on  the  first  of  these  two 
days,  B,  its  place  on  the  second,  CD  a  portion  of  the  equator,  P 
its  north  pole,  and  AB  a  portion  of  the  ecliptic.  Then  will  E  he 
the  place  of  the  vernal  equinox. 

The  arcs  PA  and  PB  are  known;  and  from  the  interval  in  si¬ 
dereal  time  between  the  two  observations,  the  angle  APB  or  its 
measure  the  arc  CD,  is  likewise  known,  it  being  evidently  equal 
to  the  excess  of  the  interval  above  24  hours,  converted  into  de¬ 
grees  or  parts  of  a  degree.  But, 

tan  AC  =  tan  E  sin  EC, 
tan  BD  ==  tan  E  sin  ED. 
therefore  tan  AC  :  tan  BD  :  :  sin  EC  :  sin  ED, 
or  tan  AC  -f  tan  BD  :  tan  AC  —  tan  BD  :  :  sin  EC  -f  sin 
ED  :  sin  EC  —  sin  ED. 

From  whence  we  have  (App.  32), 
sin  (AC  +  BD)  :  sin  (AC  —  BD)  :  :  tan  (EC  +  ED)  : 
tan  A  (EC  —  ED), 

or  tan  A  (EC  —  ED)  =  tan?fEC  4-  ED)  sin  (AC  —  BD) 
2  v  '  sin  (AC  +  BD) 

tan  l  CD  sin  (PA  —  90°  —  90°  -j-  PB) 
sin  (PA  —  90°  +  90°  —  PB) 
tan  ■}  CD  sin  (PA  -f  PB  —  180°) 
sin  (PA  —  PB) 

Now  knowing  CD  and  ^  (EC  —  ED),  we  know  EC  and  ED, 
which  are  the  sun’s  distances  from  the  equinox  at  the  times  of 
observation.  The  sun’s  motion  in  right  ascension  during  a  day 
may  be  considered  uniform,  particularly  near  the  equinox,  as  may 
be  determined  by  observing  it  for  several  days  about  that  time.  If, 
therefore,  CE  converted  into  time  (3.6),  be  added  to  the  side¬ 
real  time  of  the  first  observation,  we  shall  have  the  sidereal  time 
of  the  sun  being  at  the  equinox. 

7*  By  similar  observations  made  the  ensuing  year, 
the  time  of  the  sun’s  return  to  the  vernal  equinox,  will 
be  known. 

8.  The  interval  of  time  between  two  consecutive  re- 
8 


50 


ASTRONOMY. 


turns  of  the  sun  to  the  vernal  equinox  is  called  a 
Tropical  Year. 

The  ancient  astronomers  determined  the  length  of 
the  year  from  the  sun’s  return  to  the  same  tropic  and 
thence  applied  to  it  the  term  tropical  year,  which  is 
still  retained. 

9.  The  length  of  the  tropical  year  is  subject  to  a 
slight  variation.  By  observations  made  at  intervals  of 
50,  60  or  100  years,  its  mean  length,  expressed  in 
mean  solar  time  is  found  to  be  365  d.  5  h.  48  m. 
51.6  sec. 

Hence  365  d.  5  h.  48  m.  51.6  sec. :  1  day  :  :  360°  : 
59'  8."  33  =  sun’s  mean  motion  in  longitude  during  a 
mean  solar  day. 

10.  On  account  of  the  annual  precession  of  the 
equinox  iu  longitude,  which  is  50."1  (6.33),  the  sun 
only  passes  through  an  arc  of  the  ecliptic  equal  to  359° 
59'  9".9,  during  a  tropical  year. 

11.  The  time  in  which  the  sun  passes  through  the 
whole  360°  of  the  ecliptic,  or  which  is  the  same  thing, 
the  interval  of  time  between  two  consecutive  returns  of 
the  sun  to  the  same  fixed  star,  is  called  a  Sidereal 
Year . 

Hence  359°  59'  9".9  :  360°  :  :  365  d.  5  h.  48  in. 
51.6  sec.  :  365  d.  6  h.  9  m.  11.5  sec.  =  the  length  of 
a  sidereal  year,  expressed  in  mean  solar  time.  The 
sidereal  year  therefore  exceeds  the  solar,  by  20  m. 
19.9  sec. 

12.  When  the  sun’s  apparent  diameter  is  accurately 
observed,  at  different  seasons  in  the  year,  it  is  fouud  to 
vary.  It  is  greatest  about  the  first  of  January  and  con¬ 
tinually  decreases  till  about  the  first  of  July,  when  it  is 
least.  It  then  increases  till  the  first  of  January. 
When  greatest  it  is  32'  35". 6,  and  when  least,  it 


CHAPTER  VII. 


51 


is  31'  31".  Consequently  the  mean  diameter  is  32' 
3"  3.  As  there  is  no  reason  to  suppose  that  this  change 
in  the  apparent  diameter,  is  caused  by  a  change  in  its 
real  diameter,  it  is  inferred  that  the  sun’s  distance  from 
the  earth  is  variable. 

13.  From  a  comparison  of  the  sun’s  apparent  diame¬ 
ter,  as  observed  at  any  two  different  times,  we  may  ob¬ 
tain  the  ratio  of  its  distances  from  the  earth  at  those 
times.  Let  AB  and  A'B',  Fig .  13,  be  the  sun  in  two 
different  situations,  and  E  the  place  of  the  earth. 

Put  $  =  apparent  diameter  AEB 
=  apparent  diameter  A'EB' 

D  =  ES  and  D'  =  ES'. 

Then  D  sin  *  *  =  AS  =  A'S'  =  D  sin  a  <?', 
orD  =  sin  j  ¥  =  _  * 

D'  sin  |  ^  £  <?  $  * 

14).  The  apparent  diameter  of  the  sun,  when  at  his 
mean  distance  from  the  earth,  is  32'  2"-7. 

In  the  same  manner  as  for  the  moon’s  parallax  (5.15),  if  ^ 
and  be  the  sun’s  apparent  diameter,  at  its  greatest,  least,  and 
mean  distances,  we  shall  have, 


15.  From  the  sun’s  apparent  diameter  and  horizontal 
parallax,  its  real  diameter  may  be  determined.  It  is 
about  110  times  the  earth’s  diameter. 

If  d  =  2  AS  =  sun’s  real  diameter,  we  have, 

d  =  2  D  sin  A  =  (5.6)  2  R  sini.i  =  R  ?))!-£  =  R.  i 

sin  Sr  sin  -a-  w 

i  7 

—  R  Q— -  A-  =  R  x  221  =  110£  times  the  earth’s  diameter, 
o  .7 


52 


ASTRONOMY. 


10.  The  sun’s  right  ascension  may  be  obtained  by 
observing  the  sidereal  time  of  its  passage  over  the  me¬ 
ridian.  From  the  right  ascension,  the  longitude  may 
be  calculated  (6.19).  The  sun’s  longitude  thus  obtain¬ 
ed  at  different  times  in  tire  year,  does  not  increase  uni¬ 
formly  with  the  time.  Its  greatest  motion  in  longitude 
during  a  mean  solar  day,  is  61'  10"  and  takes  place  at 
the  time  its  apparent  diameter  is  greatest.  Its  least  mo¬ 
tion  is  57'  11">  and  takes  place  when  the  apparent 
diameter  is  least. 

17-  The  curve  which  the  sun’s  centre  seems  to  de, 
scribe  in  the  plane  of  the  ecliptic,  during  a  year,  is 
called  the  Apparent  Orbit  of  the  sun. 

18.  A  right  line,  conceived  to  he  drawn  from  the 
centre  of  the  sun  to  the  centre  of  the  earth,  or  to  the 
centre  of  a  planet,  is  called  the  Radius  Vector  of  the 
earth  or  planet.  A  right  line  joining  the  centres  of  the 
earth  and  moon,  is  called  the  radius  vector  of  the  moon. 

19.  It  appears  from  the  change  in  distance  between 
the  sun  and  earth  (IS),  that  the  sun’s  apparent  orbit  is 
not  a  circle;  or  at  least  that  the  earth  does  not  occupy 
the  centre. 

Let  ADBF,  Fig.  14,  be  the  apparent  orbit  of  the 
sun,  E  the  earth’s  place,  A  the  sun’s  place  wiien  the 
apparent  diameter  is  least,  B  its  place  w  hen  the  appa¬ 
rent  diameter  is  greatest,  and  I)  its  place  at  some  other 
time.  The  difference  between  the  sun’s  longitudes  at  A 
and  D,  that  is  when  the  apparent  diameters  are  least 
and  greatest,  is  found  to  be  180°.  It  follows  therefore 
that  EA  and  EB  must  form  one  straight  line  AB.  If 
AB  he  bisected  in  C,  then  AC  =  §  AB  =  |  (AE  4. 
EB). 

The  angle  AED  =  sun’s  long,  at  D  —  sun’s  long, 
at  A. 


CHAPTER  VII. 


53 


20.  The  apparent  orbit  of  the  sun,  is  an  ellipse, 
having  the  earth  in  one  focus.  This  fact  was  discover¬ 
ed  by  Kepler,  and  it  is  called  Kepler’s  first  Law.  It 
is  deduced  from  investigations,  founded  on  the  observ¬ 
ed  apparent  diameter  of  the  sun  at  different  longitudes. 

Put  ^  =  sun’s  apparent  diameter  at  A, 

V  =  do.  B, 

=  do.  D, 

m  —  J-  (P  +  f)  =  sun’s  mean  apparent  diameter, 
and  n  =  \  ($'  —  ^). 

From  the  sun’s  longitude  and  apparent  diameter,  as  obtained 
at  different  times  in  the  year,  it  is  found  that  whatever  be  the 
situation  of  D,  we  have  <F  =  m  —  n  cos  AED. 


But  (13)  AE  :  EB  :  : 

AE  :  AE  +  EB  :  :  ^ 

AE  :  i  (AE  -f  EB)  :  :  *  :  |  +  J), 

AE  :  AC  :  :  P  :  m  =  P. 


Again,  AE  :  EB  :  :  S'  :  S, 

AE  +  EB  :  AE  —  EB  :  :  P  +  S  :  V  —  S, 
2  AC  :  2  EC  : :  *  (S'  +  S)  :  A  (S'  —  3), 


or  AC  :  EC  :  :  m  :  n  =  m  = 

AO 


EC  AC^,_EC 
AC'  AE  “  AE  °  ' 


Hence  substituting  the  values  of  m  and  w,  in  m  —  n  cos  AED, 
we  have, 


rp 

But  (13)  r  =  S';  therefore, 


ASTRONOMY. 


51 

ED.  (AE  —  EC  cos  AED)  =  AE.  EB  =  (AC  +  EC). 
(AC  —  EC)  =  AC2  —  EC2, 

p  AC2  —  EC2 

°f  _  AC  —  EC  cos  AED' 

But  (Conic  Sections*)  the  last  equation  expresses  a  property 
of  an  ellipse,  of  which  AB  is  the  transverse  axis,  C  the  centre, 
and  E  one  of  the  foci. 

21.  The  point  which  is  the  sun’s  place,  when  most 
distant  from  the  earth,  is  called  the  Apogee;  and  the 
point  which  is  its  place  when  nearest  the  earth,  is  call¬ 
ed  the  Perigee.  Those  points  are  also  called  Apsides ; 
the  most  distant  being  called  the  Higher  Apsis ,  and  the 
nearest,  the  Lower  Apsis.  The  transverse  axis,  which 
joins  the  apsides,  is  called  the  Line  of  the  Apsides . 

22.  The  distance  between  the  earth  and  the  centre 
of  the  sun’s  apparent  orbit,  is  called  the  Eccentricity 
of  the  orbit. 

23.  Astronomers  usually  call  the  mean  distance  of 
the  earth  from  the  sun,  a  unit  or  1,  and  express  other 
distances  in  conformity  with  this  assumption. 

21.  Considering  the  earth’s  mean  distance  from  the 
sun,  equal  1,  the  eccentricity  of  the  sun’s  apparent  orbit 
is  .0168  nearly. 

Put  AC  =  1,  and  e  =  EC  =  the  eccentricity;  then  (20), 
EC  e 

n  =  m  =  -  m  =  em, 
n  32". 3 

0r  £  =  m  =  1923*73  =  '°16S  Uear,-V- 

25.  Let  ADEP,  Fig.  15 ,  be  the  sun’s  apparent  orbit, 
E  the  place  of  the  earth,  D  the  sun’s  place  at  any  time, 

*  The  proposition  here  referred  to,  though  an  important  one,  is  omitted 
in  several  of  our  treatises  on  Conic  Sections.  It  is  therefore  demonstrated 
in  the  appendix,  article  51. 


CHAPTER  VII. 


55 


and  A  and  B  the  apogee  and  perigee.  Then  by  the 
apparent  motion  of  the  sun,  the  radius  vector  ED, 
moves  about  E,  in  such  manner  that  the  area  of  the 
sector  AED,  increases  uniformly  with  the  time.  This 
fact,  discovered  by  Kepler,  is  usually  expressed  by 
saying,  the  Radius  Vector  describes  equal  Jlreas  in 
equal  Times.  It  is  called  Kepler’s  Second  Law. 


If  the  circle  AGB  be  described  on  the  transverse  axis  AB,  and 
IIDG  be  drawn  perpendicular  to  AB,  then  (Conic  Sections*), 


AC2  —  EC: 


AC  —  EC  cos  AED 
+  EC  cos  ACG; 

hence,  EC  cos  ACG  = 


=  ED  =  AC  —  EC  cos  BCG  =  AC 


AC: 


EC 


EC  cos  AED 


—  AC 


cos  AED 


AC 

AC.  EC  cos  AED  —  EC2 
AC  —  EC  cos  AED  ’ 

»  pp  AC  cos  AED — EC  — & 

or  cos  ACG  =  — — — — — - = - f 

AC  —  EC  cos  AED  1  —  e  cos  AED 

Now  the  area  of  ECG  =  j-  EC.  CG  sin  ACG  =  §  e  sin  ACG, 

area  of  ACG  =  AC.  A  arc  AG  =  |  arc  AG, 

area  of  AEG  =  $  (e  sin  ACG  +  arc  AG)  (B). 

Hence  the  area  AEDA  =  55.  AEG  =  —  .  (e  sin 

ACG  -|-  arc  AG) 


The  eccentricity  e  being  known  (24),  and  also  the  angle  AED, 
from  the  sun’s  longitudes  at  A  and  D  (19),  the  angle  ACG  or  arc 
AG  becomes  known  (A);  and  thence  (C)  the  area  of  the  ellipti¬ 
cal  sector  AEDA.  The  area  of  the  sector  AEDA,  thus  obtained  at 
different  times  in  the  year,  is  found  to  increase  uniformly  with  the 
time. 


26.  If  we  suppose  the  sun  to  be  situated  at  E,  and 
the  earth  to  revolve  round  it  in  the  orbit  ADBP,  and 


*  See  Appendix,  article  52,. 


56 


ASTRONOMY. 


if  we  admit  the  distances  of  the  fixed  stars  to  be  so 
great,  that  straight  lines  conceived  to  be  drawn  from 
the  points  Til  and  A  to  meet  at  any  one  of  them,  will 
not  contain  an  appreciable  angle,  or  which  amounts  to 
the  same  thing,  that  these  lines  may  be  considered  as 
sensibly  parallel;  then  the  sun  will  appear  to  move  ex¬ 
actly  in  the  same  manner  as  on  the  supposition  that  it 
is  really  in  motion  about  the  earth  at  rest  at  E. 

Let  us  suppose  some  one  of  the  fixed  stars  to  coin¬ 
cide  with  the  vernal  equinox,  and  let  EQ  and  TAQ', 
be  two  straight  lines  which  being  produced  would 
meet  at  this  star.  Then  on  the  supposition  that  the 
earth  is  at  rest  at  E,  QEA  is  the  sun’s  longitude  at  A, 
and  QED  is  its  longitude  at  D.  Therefore  AED  = 
QED  —  QEA,  is  the  difference  of  their  longitudes  at 
A  and  D. 

On  the  supposition  that  the  sun  is  at  rest  at  E,  and 
considering  EQ  and  TQ'  as  parallel,  its  longitude 
when  the  earth  is  at  A,  is  180°  4-  EAT  =  180°  4- 
QEA.  In  like  manner  its  longitude,  when  the  earth 
is  at  D,  is  180°  4-  QED.  But  180°  4-  QED  — 
(180°  +  QEA)  ==  QED  —  QEA  =  AED.  The 
difference  of  its  longitudes,  and  consequently  its  appa¬ 
rent  motion,  is  therefore  the  same  on  the  two  suppo¬ 
sitions. 

But  when  we  consider  that  the  sun’s  diameter  is 
more  than  a  hundred  times  the  diameter  of  the  earth 
(15),  and  consequently  its  magnitude,  more  than  a  mil¬ 
lion  times  the  magnitude  of  the  earth,  it  seems  more 
reasonable  to  suppose  that  the  sun’s  apparent  motion 
is  produced  by  a  real  motion  of  the  earth,  than  to  adopt 
the  contrary  supposition. 

37*  The  apparent  diurnal  motion  of  the  heavenly 
bodies  from  east  to  west,  may  likewise  be  accounted 


CHAPTER  VII. 


37 


for,  by  admitting  the  earth  to  have  a  diurnal  motion  on 
ifs  axis  in  a  contrary  direction,  that  is  from  west  to 
east.  And  it  is  certainly  more  reasonable  to  suppose 
this  diurnal  motion  of  the  earth,  than  to  suppose  that 
all  the  heavenly  bodies,  situated  at  various  and  im¬ 
mensely  great  distances,  should  have  motions  so  ad¬ 
justed  as  to  revolve  round  the  earth  in  the  same  length 
of  time. 

Various  astronomical  phenomena  serve  to  prove  that 
the  earth  really  has  these  two  motions;  that  is,  an  an¬ 
nual  motion  round  the  sun,  and  a  diurnal  motion  on 
its  axis. 

During  the  annual  motion  the  earth’s  axis  continues 
parallel  to  itself;  or  in  other  words,  if  we  suppose  a 
right  line  to  remain  fixed  in  the  position  which  the 
earth’s  axis  has  in  one  part  of  the  orbit,  the  axis  con¬ 
tinues  during  the  whole  annual  revolution,  nearly  pa¬ 
rallel  to  that  line. 

As  the  earth’s  axis  is  perpendicular  to  the  plane  of 
the  equator,  the  angle  contained  between  it,  and  a  right 
line  passing  through  the  centre  of  the  earth  perpendicu¬ 
lar  to  the  ecliptic,  must  be  equal  to  the  angle  contained 
by  these  planes,  that  is  to  the  obliquity  of  the  ecliptic. 
The  line  perpendicular  to  the  plane  of  the  ecliptic  is 
called  the  Axis  of  the  Ecliptic. 

88.  Although  the  particular  consideration  of  the 
planets,  is  referred  to  a  succeeding  part  of  the  work,  it 
may  be  here  observed  that  all  of  them,  including  the 
earth  as  one,  revolve  round  the  sun,  from  west  to  east, 
at  different  distances  and  in  different  times;  and  that 
the  moon  revolves  round  the  earth,  and  with  it  round 
the  sun.  This  system  of  the  sun  and  planets  is  called 
the  Coper nican  System ,  from  its  inventor  Copernicus. 

The  order  of  the  planets  with  respect  to  their  dis* 

9 


58 


ASTRONOMY. 


tances  from  the  sun,  is  Mercury,  Venus,  the  Earth , 
Mars,  Vesta ,  Juno ,  Ceres ,  Valias,  Jupiter,  Saturn 
and  Uranus . 

29.  The  same  reasoning  that  has  been  used  to  prove 
that  the  sun’s  apparent  orbit  is  an  ellipse  and  that  the 
radius  vector  describes  equal  areas  in  equal  times,  ap¬ 
plies,  when  we  suppose  the  sun  at  rest,  to  prove  that 
the  earth’s  orbit  is  an  ellipse,  having  the  sun  in  one 
focus,  and  that  its  radius  vector  describes  equal  areas 
in  equal  times. 

Kepler,  extending  his  researches,  found  that  the  or¬ 
bits  of  the  planets  are  ellipses  and  that  the  radius  vec¬ 
tor  of  each  describes  equal  areas  in  equal  times. 

30.  Another  important  law,  discovered  by  Kepler, 
is,  that  the  square  of  the  time  in  which  any  planet  re¬ 
volves  round  the  sun,  is  to  the  square  of  the  time  in 
which  another  planet  does  the  same,  as  the  cube  of 
the  mean  distance  of  the  former,  from  the  sun,  is  to  the 
cube  of  the  mean  distance  of  the  latter.  This  relation, 
which  is  usually  expressed  by  saying  the  Squares  of 
the  times  of  revolution  of  the  Planets  are  as  the  Cubes 
of  their  mean  distances  from  the  Sun ,  is  called  Kep¬ 
lers  Third  Law. 

31.  Although  astronomers  have  completely  estab 
lished  the  fact  of  the  earth’s  annual  and  diurnal  mo¬ 
tions,  yet  with  a  view  to  convenience  of  expression, 
they  still  frequently  speak  of  the  sun’s  orbit  and  of  the 
motion  of  the  sun. 

32.  The  point  in  the  earth’s  or  a  planet’s  orbit, 
which  is  the  most  distant  from  the  sun,  is  called  the 
Aphelion;  and  the  nearest  point  is  called  the  Perihe¬ 
lion.  The  terms  apogee  and  perigee  are  only  used  to 
express  the  greatest  and  least  distances  from  the  earth, 

33.  Let  D  and  F,  Fig.  14,  be  two  situations  of  the 


CHAPTER  VII. 


39 


sun  in  its  apparent  orbit,  at  which  its  apparent  diame¬ 
ter  is  the  same.  Then  ED  =  EF,  and  consequently 
the  angle  AED  =  AEF.  Hence  from  the  longitudes 
of  the  sun  at  F  and  D,  the  longitude  of  A,  the  apogee, 
becomes  known.  The  following  is  however  a  more  ac¬ 
curate  method  of  determining  the  place  of  the  apogee. 

The  transverse  axis  AB  divides  the  ellipse  into  two 
equal  parts.  The  sun  will  therefore  be  as  long  in 
moving  from  A  to  B  as  from  B  to  A  (23).  Hence  it 
will  be  half  a  year  in  moving  from  A  to  B,  and  in  this 
time  it  will  change  its  longitude  180°.  This  will  not 
be  the  case  for  any  other  points  in  the  orbit,  because  no 
other  right  line,  passing  through  E,  divides  the  ellipse 
into  equal  parts.  If  therefore  two  longitudes  of  the 
sun  be  observed  at  the  interval  of  half  a  year,  and  such 
that  their  difference  is  180°,  the  position  of  the  line  of 
the  apsides  becomes  known. 

We  have  supposed  the  position  of  the  line  of  the 
apsides  to  remain  fixed.  But  observations  made  at 
long  intervals  of  time,  prove  that  it  has  a  slow  motion 
in  the  order  of  the  signs.  Astronomers  therefore  mo¬ 
dify  the  preceding  methods  of  determining  the  position 
of  the  apsides,  so  as  to  take  notice  of  this  motion,  which 
however  only  amounts  to  a  few  seconds  in  a  year. 

34.  Delambre  determined  the  longitude  of  the  apogee, 
for  the  beginning  of  the  year  1800  to  be  3s  9°  29'  3" 
and  its  mean  yearly  increase  in  longitude  to  be  61". 9. 

33.  If  we  divide  3s  9°  29'  3"  by  61". 9,  the  quotient 
is  3786.  Hence  it  appears  that  about  3786  years  an¬ 
terior  to  the  year  1800,  the  apogee  coincided  with  the 
vernal  equinox.  It  is  worthy  of  remark,  that  it  is  about 
that  period,  at  which  chronologists  generally  fix  the 
creation  of  the  world. 

36.  If,  from  6l".9  the  annual  motion  of  the  apogee 


60 


AS  T  RON  0 M  Y • 


from  tlie  equinox,  we  subtract  50".  1,  the  annual  preces¬ 
sion  of  the  equinoxes,  tiie  remainder  which  is  11  "JS,  is 
the  mean  annual  motion  of  the  apogee. 

37.  The  time  between  two  consecutive  returns  of 
the  sun  to  the  apogee,  is  called  an  Anomalistic  Tear. 

As  359°  50'  9". 9  :  360°  0'  11".8  :  :  365d.  5 h.  48m. 
51.6  sec. :  365  d.  6  h.  13  m.  58.8  sec.  =  the  length  of  the 
anomalistic  year,  expressed  in  mean  solar  time.  The 
anomalistic  year,  therefore,  exceeds  the  tropical,  by 
25  m.  7-2  sec. 

As  365  d.  6  h.  13  m.  58.8  sec.  :  1  day  :  :  360°  :  59' 
g". 16  =  sun’s  mean  motion  from  the  apogee  during  a 
mean  solar  day. 

38.  At  the  time  that  the  true  place  of  the  sun  is  at  J), 
Fig.  15,  let  F  be  the  place  at  which  it  would  have 
been,  if  it  had  moved  from  the  apogee  with  its  mean 
angular  motion  of  59'  8".  16  a  day. 

39.  The  angle,  contained  between  the  line  of  the 
apsides  and  the  radius  vector,  is  called  the  True 
Anomaly .  Thus  AED  is  the  true  anomaly. 

40.  The  angle,  contained  between  the  line  of  the 
apsides  and  the  straight  line  from  the  earth  to  the  mean 
place  of  the  sun,  is  called  the  Mean  Anomaly.  Thus 
AEF  is  the  mean  anomaly. 

The  angle  ACG  is  called  the  Eccentric  Anomaly. 

41.  The  angle,  which  is  the  difference  between  the 
Mean  and  True  anomalies,  is  called  the  Equation  of 
the  Centre. 

42.  The  equation  of  the  centre  DEF,  expresses  the 
difference  between  the  mean  longitude  QEF  and  the 
true  longitude  QED.  The  mean  longitude  and  mean 
anomaly  increase  uniformly  with  the  time.  They  may, 
therefore,  be  easily  determined  for  any  particular  point 
of  time.  Then,  if  the  equation  of  the  centre,  corres. 


CHAPTER  VII. 


61 


ponding  to  the  mean  anomaly  be  known,  the  true  lon¬ 
gitude  becomes  also  known. 

}  /  43.  The  problem  for  determining  the  true  anomaly 
from  the  mean,  and  which  therefore  determines  the 
equation  of  the  centre,  is  called  JCejrtcr’s  Problem.  It 
is  a  problem  of  great  importance  and  has  been  solved 
in  various  ways.  The  following  method  combines 
simplicity  with  a  requisite  degree  of  accuracy. 


Let  CL  be  drawn  parallel  to  EF,  LO  parallel  to  GC,  and  EM 
perpendicular  to  GC  produced.  Put, 

T  =  time  of  describing  the  whole  ellipse, 
t  —  time  of  describing  AD, 
e  —  EC,  AC  being  =  1, 
u  —  ang.  AED  ==  true  anomaly, 

x  =  arc  AG  =  ang.  ACG  ==  eccentric  anomaly,  /- 

z  =  arc  AL  =  ang.  ABL  =  ang.  AEF  =  mean  anomaly,  * 
M  =  area  of  the  circle  AGB, 

N  =  area  of  the  ellipse  ADBP, 
p  =  6.28318  &c.  =  circumference  of  the  circle. 

Then  M  =  AC  x  arc  AGB  =  1  x 

Now  (Conic  Sections) 


M  :  N  :  :  AC  :  CR  : :  AEGA  :  AEDA, 

M  :  AEGA  : :  N  :  AEDA  : :  T  :  t :  :  3603 :  ang.  AEF  : 

hence  AEGA  =  ^  2  =  if .  2 
P  P 

But  (25.B)  AEGA  =  \  (e  sin  x  4.  x). 

Therefore  z  =  x  +  e  sin  x 


i  *; 

V- 


:p:z; 


(D), 


Again  (25.A)  cos  x  =- 


cos  u — e 


1  —  e  cos  u 
cos  x  —  e  cos  u  cos  x  =  cos  u  —  e, 
cos  u  -f  e  cos  u  cos  x  =  e  -f  cos  x, 

cos  u  =  e  +  cos  * 

1  -f  e  cos  x 


(E). 


The  last  formula  may  be  converted  into  another  that  will 
be  more  convenient  for  logarithmic  computation.  We  have 
(App.12.), 


63 


ASTRONOMY. 


tan  2  \  u 


1  —  cos  M 
1  4-  cos  u 


1  +  e  cos  x  —  e  —  cos  a; 
1  4-  e  cos  x  4-  e  4-  cos  x 
(1  —  c)  —  (1  —  e)  cos  x 
“  (1  4  e)  4  (1  4  c)  cos  x 

1.~c  tan2  ix. 

1  +  e 


1  — 


1  + 


t  4-  cos  x 
1  4-  ecosa; 
e  4-  cos  x 
1  4-  ecorx 


1  —  e  1  —  cos  x 
1  4-  e  i  4-  cos  x 


Hence  tan  \  u  =  tan  J 


£  x/ 


1 

1 


We  have  now  an  expression  for  the  true  anomaly  in  terms  of 
the  eccentric,  and  (D)  an  equation  showing  a  relation  between 
the  mean  and  eccentric  anomalies.  But  in  consequence  of  the 
latter  containing  both  x  and  sin  x ,  we  can  not,  in  a  direct  manner, 
obtain  the  eccentric  from  the  mean  anomaly.  There  are  how¬ 
ever  various  methods  of  approximation  which  give  its  value  to 
any  required  degree  of  accuracy. 

44.  We  have  by  trigonometry, 

EM  =  EC  sin  ECM  =  EC  sin  ACG  =  e  sin  *; 
but  (43. D)  e  sin  x  =  z  —  x  =  arc  AL  —  arc  AG  =  arc  LG. 
Therefore  EM  =  arc  LG. 

Because  OL  is  parallel  to  MG,  and  EM  is  perpendicular  to  it, 
OM  =  sin  LG.  The  difference  between  EM  and  OM  is  there¬ 
fore  equal  to  the  difference  between  the  arch  LG  and  its  sine; 
and  in  all  cases  for  the  sun  and  planets,  this  difference  is  small, 
as  may  be  thus  shown.  Since  LG  =  e  sin  x ,  it  is  evident  that 
LG  will  be  greatest  when  x  =  90°,  and  then  we  have  LG  ==  e 
sin  90°  =  e.  But  for  the  sun’s  apparent  orbit  e  —  .0168  nearly 
(24);  therefore  LG  =  .0168  =  0°  5T'|.  The  value  of  e  for  the 
most  eccentric  orbit  of  the  planets  is  about  .254,  the  mean  dis¬ 
tance  from  the  sun  being  expressed  by  1 .  Consequently  LG 
when  greatest  is  about  14°-|. 

The  difference  between  an  arc  of  14°|  and  its  sine,  is  but  . lit¬ 
tle,  and  for  an  arc  of  57'f  it  is  much  less.  It  follows  therefore 
that  in  all  cases,  OM  is  nearly  equal  to  the  arc  LG  or  its  equal 
the  right  line  EM.  Consequently,  because  OL  is  parallel  to  CG, 
EL  is  nearly  parallel  to  it,  and  the  angle  AEL  is  nearly  equal  to 
the  eccentric  anomaly  ACG. 


63 


CHAPTER  VI. 


Now  2  =  ACL  =  CEL  -f  CLE, 

Put  0  =  CEL  —  CLE, 
w  =  CEL, 

y  =  ACG  —  CEL  =  x  —  w. 

Then  a;  =  w  -f-  y. 

By  trigonometry, 

CL  4.  EC  :  CL  —  EC  :  :  tan  f  (CEL  +  CLE)  :  tan  i 
(CEL  —  CLE), 


or  1  -f  e  :  1  — e  :  :  tan  \  z  :  tan  J  0. 

Therefore  tan  ^  0  —  T—  tan  4  z, 

2  1  +  e 

w  —  CEL  =  \  z  0- 

Now  (43.  D)  and  (App.  13), 

r  =  x  -j-  e  sin  x  —  w  -\-  y  -f  e  sin  (to  +  y)  =  to  +  V  +  e 
sin  w  cos  y  -f  e  cos  w  sin  y. 

But  because  y  is  very  small  we  may  take  gos  y  =  1  and 
sin  y  =  y. 

Then  z  =  w  +  y  +  e  sin  w  -f  e  y  cos  w, 
y  4-  e  y  cos  to  =  2:  —  (w  4-  e  sin  w), 
z  —  (w  4.  e  sin  w)  m 
^  1  4-  e  cos  w 

and  x  =  w  4- 1/,  becomes  then  known. 

If  greater  accuracy  is  required,  put  w  4-  y  =  a',  and  y'  ==■ 
a;  —  a;'.  Then  reasoning  as  above  we  have; 

,  z  —  (x'  4-  e  sin  xr 
1  4-  e  cos  x' 
and  x  a=  x'  4-  y\ 

The  last  formula  is  not  necessary  except  very  great  accuracy 
is  required.  The  value  of  x  =  w  4-  ?/,  is  not  liable  to  a  greater 
error  than  TlT  of  a  second  for  the  most  eccentric  orbit  of  the  pla¬ 
nets.  For  the  sun,  the  error  can  not  amount  to  more  than  about 
of  a  second  if  we  take  x  =  w. 


When  the  value  of  x  is  found,  we  obtain  the  true  anomaly 
from  the  expression 

tan  a  u  =  tan  |  x  */  \ZZ1. 

2  2  1  4-e 

45.  Some  astronomers  have  proposed  that  the  anomaly  should 
always  be  reckoned  from  the  perigee  because  it  is  necessary  to  do 
so  for  the  orbits  of  comets. 


64 


ASTRONOMY. 


If  180°  be  added  to  the  mean  anomaly,  reckoned  from  the 
perigee,  rejecting  360°  when  the  sum  exceeds  it,  the  result  will 
be  the  mean  anomaly  reckoned  from  the  apogee.  Then  the  true 
anomaly  from  the  apogee  may  be  found  by  the  preceding  formulae. 
The  difference  between  these  will  be  the  equation  of  the  centre. 

EXAMPLE  I. 

Given  the  sun’s  mean  anomaly  from  the  perigee  89  25°  or 
from  the  apogee  2s  25°,  and  the  excentricity  of  the  orbit  .016774; 
required  the  equation  of  the  centre. 

Here  z  =  2s  25°  =  85% 
and  e  =  .016774. 


Log.  (1— e)  .983226 

9.9926533 

log.  (1  +  e)  1.016774  - 

0.0072244 

log.  *  c 

tan  \  z  42°  30' 

9.9854289 

9.9620525 

tan  i  9  -  41  32  38".6  - 

9.9474814 

w  =  -  84  2  38.6 

*  s  =  *  w  =  42  1  19.3 

- 

t  1— « 

Log.  x—  -  -  -  - 

9.9854289 

1  y  1  — e 

logy--  -  -  - 

tan**  42°  1'  19".3  - 

9.9927144 

9.9547732 

tan*  u  41°  32'  40" 

9.9474876 

u  =  83  5  20 

i.  85  0  0 

it  —  z  =  —  1  54  40  =  equation  of  sun’s  centre.- 


CHAPTER  VII, 


65 


EXAMPLE  II. 


Given  Mercury’s  mean  anomaly  from  the  aphelion  l9  28%  and 
the  eccentricity  of  its  orbit  .205513;  required  the  equation  of  the 
centre. 


2  — 

583 

and  e  =  .205513. 

Log.  (1  —  e) 

- 

.794487  - 

9.9000868 

log.  (1  +  e)  - 

1.205513 

0.081X719 

1  1  — e 

log.- 

1  +  e 

- 

- 

9.8189149 

tan  A  z 

29°  0'  0"  - 

9.7437520 

tan  ^  6 

- 

1 

0 

9.5626669 

w  = 

49  4  4.7 

Log.  c-—  .205513 

- 

-  9.3128393 

log.  -  20G264".8 

- 

- 

5  3144251 

sin  ip  49°  4'  4".7 

- 

-  9.8782272 

e  sinie  =  32025".2  = 

8" 

53'  45".2  - 

4.5054916 

to=- 

49 

4  4.7 

4.  e  sin  iu  = 

57 

57  49.9 

z  = 

58 

0  0 

z  —  (w  4-  e  sin  w) 

0 

2  10.1  =  130".l 

log.  e  = 

.205513 

9.3128393 

cos  to 

49  4'  4". 7  - 

-  9.8163495 

t  cos  w 

.1346 

9.1291888 

1  -4  e  cos  w 

1.1346 

130"! 
y  ~  0346 

==  11 4". 7  =1'  54".7 

x  =  w  4.  y  =  49  4'  4".  7  i-  V  54". 7  ~  49J  5'  59".  4 
10 


-  9.8189149 


9.909  4574 
9.6591633 

w  =  -  40  41  29.8 

*  =  -  58  0  0 

u  —  z  — —  17  18  30.2  ==  equation  of  the  centre. 

46.  When  the  eccentricity  of  the  orbit  is  known, 
the  true  radius  vector,  corresponding  to  any  given  mean 
anomaly,  may  thence  be  determined. 

Put  r  =  ED  =  the  radius  vector;  then  because  (25), 

ED  =  AC  +  EC  cos  ACG,  we  have, 
r  —  1  +  e  cos  x. 

Hence,  having  computed  the  eccentric  anomaly,  corresponding 
to  the  given  mean  anomaly,  we  easily  obtain  the  value  of  the  ra¬ 
dius  vector. 

47.  If  we  divide  360°  by  the  number  of  hours  in  the 
time  of  revolution,  of  the  sun  or  a  planet,  in  its  orbit, 
the  result  will  be  the  mean  hourly  motion.  If  H  be  the 
mean  hourly  motion,  H'  the  true  hourly  motion,  e  the 
eccentricity  of  the  orbit,  and  r  the  radius  vector,  then 
we  have  for  the  true  hourly  motion,  the  following  ex- 
pression. 

H'  =  H.  ^  ^  —  e 
r2 

Let  a  and  b  be  the  places  of  the  body,  half  an  hour  before  and 
after  it  is  at  P.  Then  will  the  angle  aE5,  be  its  true  hourly  mo¬ 
tion  in  this  part  of  its  orbit.  With  the  centre  E  and  radius  EP 
describe  the  arc  cd .  Then  since  the  angle  aEb  is  very  small,  we 


66 


ASTRONOMY. 


l0£ 


1— e 
1  -f~  e 


Ion.  ^  \ - ? 

1  +  e 


tan 

tan 


x  - 
u  - 


24°  32'59".7 
20  20  44.9 


CHAPTER  VII. 


67 


may  consider  the  elliptical  sector  aEJm,  as  equal  to  the  circular 
sector  cEdc. 

Now  the  circumference  of  the  circle  of  which  EP  is  radius,  is 
equal  EP.  p  =  rp;  and  consequently  the  area  of  the  same  circle 
is  =  r.  \rp  =  \  r2p.  Then, 

360°  :  H'  : :  r2  p  :  cEdc, 

cEdc  —  -5!-.  i  r2  p. 

360°  2  ^ 


Again  (43)  360°  :  H  : :  T  :  1  : :  N  :  abEa  : :  N  :  cEdc, 
cEdc  =  JL.  N  ==  JL.^5.M  =  7JL  v  (1—  e2) 'p. 

360°  n°  a r  qco®  ^  v  ^ 


Hence, 


IP 

360°' 


i?2  P  = 


360°  AC 
H 


360c 


v/(l 


360- 

e2)l/>, 


or  H'  =  H. 


48.  Because  H  and  v/  (i  —  e2)  are  constant  quan¬ 
tities,  it  follows  that  the  true  motion  of  a  body  in  an 
elliptical  orbit,  varies  inversely  as  the  square  of  the 
radius  vector.  Hence  it  continually  increases  from  the 
apogee,  where  it  is  least,  to  the  perigee,  where  it  is 
greatest;  and  thence  continually  decreases  to  the  apogee. 

49.  As  the  mean  and  true  places  of  the  body  coin¬ 
cide  at  the  apogee  and  perigee,  and  as  near  the  apogee 
the  true  motion  is  less  than  the  mean  motion,  the  mean 
place  will  be  in  advance  of  the  true  place,  from  the 
apogee  to  the  perigee.  From  the  perigee  to  the  apogee, 
the  true  place  will  be  in  advance  of  the  mean  place. 
It  is  therefore  evident;  that  the  equation  of  the  centre, 
which  is  the  difference  between  the  mean  and  true 
places,  and  which  at  the  apogee  is  nothing,  must  con¬ 
tinually  increase,  till  the  true  motion  becomes  equal  to 
the  mean,  when  the  equation  will  be  greatest;  and 
thence  it  will  decrease  to  the  perigee,  where  it  again 
becomes  nothing.  In  like  manner,  the  equation  of  the 


68 


ASTRONOMY. 


centre  increases  from  the  perigee,  till  the  true  motion, 
which  is  then  diminishing,  becomes  equal  to  the  mean. 
It  is  then  greatest;  and  afterwards  decreases  till  it  be¬ 
comes  nothing,  at  the  apogee. 

The  parts  of  the  orbit  on  each  side  of  the  line  of  the 
apsides,  being  symmetrical,  the  greatest  equation  on 
one  side  will  be  equal  to  the  greatest  equation  on  the 
other  side. 

The  sun’s  true  longitude  may  be  obtained  each  day 
from  its  observed  right  ascension  (6.19).  The  differ¬ 
ence  between  its  longitudes  on  any  two  consecutive 
days  will  be  its  true  diurnal  motion  at  that  time.  Hence 
we  may,  by  repeated  observations,  find  the  time  when 
the  true  diurnal  motion  is  equal  to  the  mean.  Know  ing 
then  the  meau  and  true  longitudes,  when  this  takes 
place  in  opposite  parts  of  the  orbit,  we  may  obtain  from 
thence  the  greatest  equation  of  the  centre. 

Let  A  =  the  mean  longitude,  and  B  =  the  true  lon¬ 
gitude,  at  the  time  the  motions  are  equal  between  the 
apogee  and  perigee;  A\=  the  mean  longitude,  and  B' 
=  the  true  longitude,  when  the  motions  are  equal  be¬ 
tween  the  perigee  and  apogee;  and  V  =  the  greatest 
equation  of  the  centre.  Then, 

B  =  A  —  V, 

B'  =  A'  +  y. 

Hence  B'  —  B  =  A'  —  A  +  2  V, 

2V  =  (B'  —  B)  —  (A'  —  A), 
y  =  §  (B'  —  B)  —  i  (A'  —  A). 

At  the  time  of  the  greatest  equation,  the  sun’s  true 
motion  continues  very  nearly  the  same  for  two  or  three 
days.  Consequently  the  equation  of  the  centre  will 
remain  very  nearly  the  same  during  this  time.  The. 


CHAPTER  VII. 


69 


value  of  the  greatest  equation  may  therefore  be  deter¬ 
mined  with  great  accuracy  by  this  method,  without  the 
necessity  of  knowing  very  precisely  the  time,  at  which 
the  true  motion  is  equal  to  the  mean. 

00.  In  a  preceding  article,  a  method  has  been  given 
of  obtaining  the  eccentricity  from  the  greatest  and  least 
apparent  diameters.  It  may  however  be  obtained  much 
more  accurately  from  the  greatest  equation.  Put 

lz  _  number  of  seconds  in  V 
S06264".8 


Then,  it  is  found  by  means  of  an  analytical  investi- 
igation,  which  we  shall  omit,  that 


e 


\  K 


—  Ks 
768 


087 

983040 


01.  By  a  comparison  of  observations  made  at  distant 
periods,  it  has  been  discovered  that  the  equation  of  the 
sun’s  centre,  and  consequently  the  eccentricity  of  the 
orbit,  are  at  the  present  period  continually  diminishing. 
The  rate  of  diminution  in  the  greatest  equation  is  about 
18".79  in  a  century.  It  follows,  therefore,  that  the 
equation  of  the  centre,  as  computed  for  a  given  time, 
will  not  be  accurately  true  for  a  different  time.  It  will, 
however,  err  but  little  for  a  few  years,  before  and  after 
the  time,  for  which  it  is  computed.  A  complete  table 
of  the  equation  of  the  sun’s  ce  ntre,  has  a  column  con¬ 
taining  the  variation  of  the  equation  in  a  century,  called 
the  Secular  variation,  by  means  of  which  the  correct 
equation  may  be  obtained  for  different  periods. 

52.  The  force  which  causes  heavy  bodies,  when  left 
at  liberty  near  the  surface  of  the  earth,  to  fall  to  it,  is 
called  the  Attraction  of  Gravitation .  Newton  was  led 


70 


ASTRONOMY. 


by  reasoning  which  appertains  to  Physical  Astrono¬ 
my,  to  adopt  as  a  principle,  that  this  attraction,  de¬ 
creasing  inversely  as  the  square  of  the  distance  from 
the  earth’s  centre,  extends  to  the  moon  and  retains  it, 
in  an  elliptical  orbit  about  the  earth;  that  the  sun,  moon 
and  planets  are  endued  with  like  attractive  forces, 
which  vary  according  to  the  same  law;  and  that  it  is 
the  sun’s  attraction,  which  retains  the  earth  and  planets 
in  their  orbits.  This  general  principle  of  attraction  is 
called  Newton’s  Theory  of  Universal  Gravitation. 
A  combination  of  various  discoveries  which  have  been 
made  in  astronomy  since  the  time  of  Newton,  has 
served  completely  to  establish  the  truth  of  his  theory. 

53.  If  the  earth  was  acted  on  by  no  other  force  than 
the  attraction  of  the  sun,  its  orbit  would  be  accurately 
an  ellipse,  and  the  areas  described  by  its  radius  vector 
in  equal  times,  would  be  precisely  equal.  Its  true  lon¬ 
gitude  would  therefore  be  accurately  expressed  by  its 
mean  longitude,  corrected  by  the  equation  of  the  cen¬ 
tre.  But  the  attractions  of  the  moon  aud  planets  extend 
to  the  earth,  and  some  of  them  produce  sensible,  though 
slight  effects  on  its  motion.  By  the  aid  of  very  refined 
analytical  investigations,  the  means  have  been  obtained 
of  calculating  these  effects,  which  are  called  Perturba¬ 
tions.  Their  whole  amount  may  produce  a  change  in 
the  sun’s  longitude  of  about  45";  but  in  general  it  is 
considerably  less.  Our  best  solar  tables,  which  are 
those  calculated  by  Delambre,  contain  the  equations 
due  to  the  attractions  of  the  moon  and  planets. 

The  equation  of  the  centre  and  the  amount  of  the 
perturbations,  applied  to  the  mean  longitude  of  the  sun, 
give  its  true  longitude  from  the  mean  equinox  (1). 

54.  The  difference  betw  een  the  mean  place  of  the 
equinox  in  the  ecliptic,  and  its  true  place,  is  called  the 


CHAPTER  VIII. 


71 

Equation  of  the  Equinoxes  in  Longitude,  or  sometimes, 
the  Lunar  Nutation.  The  greatest  value  of  this  equa¬ 
tion  is  about  18".  A  table,  from  which  its  value  may 
be  obtained  for  any  particular  time,  forms  a  part  of  a 
complete  set  of  solar  tables. 

To  obtain  the  sun’s  true  longitude  from  the  true 
equinox,  we  must  correct  the  mean  longitude  by  the 
equation  of  the  centre,  the  amount  of  the  perturbations, 
and  the  equation  of  the  equinoxes  in  longitude. 

55.  From  the  sun’s  true  longitude,  we  obtain  its  true 
right  ascension,  by  the  formula  in  the  last  chapter 
(6.20).  Another  method  is  by  means  of  a  table  calcu¬ 
lated  for  the  purpose. 

The  difference  between  the  longitude  and  right  as¬ 
cension,  is  called  the  1 Reduction  of  the  Ecliptic  to  the 
Equator .  Tables  have  been  calculated  which,  for  a 
given  obliquity  of  the  ecliptic,  give  the  reduction  cor¬ 
responding  to  each  degree  or  half  degree  of  longitude, 
and  also  the  variation  in  the  reduction  for  a  change  of 
T  in  the  obliquity.  With  such  a  table  the  reduction 
corresponding  to  a  given  longitude  is  easily  obtained, 
and  being  applied  to  the  longitude,  it  gives  the  right 
ascension. 

CHAPTER  VIII. 

Equation  of  Time — Right  Ascension  of  Mid- Heaven. 

1.  Solar  days,  being  determined  by  the  apparent 
diurnal  motion  of  the  sun  (7-4),  are  used  for  all  the 
eommou  purposes  of  life.  Astronomers  also  generally 
use  solar  time  except  in  determining  the  right  ascen¬ 
sions  of  bodies. 

2.  In  common  reckoning  the  day  begins  at  midnight, 
and  is  divided  into  two  portions  of  12  hours  each.  The 


72  ASTRONOMY. 

first  12  are  from  mill  night  to  noon,  anil  are  usually  de¬ 
signated  by  the  letters  A.  M.*  annexed  to  the  number 
of  the  hour.  The  latter  12,  from  noon  to  midnight, 
are  designated  by  the  letters  P.  M.* 

The  astronomical  day  begins  at  noon  of  the  com 
mon  day,  and  the  hours  are  reckoned  on  to  24.  Hence, 
any  given  time  from  noon  to  midnight  is  expressed  by 
the  same  day  and  hour  in  astronomical  and  in  common 
reckoning.  But  to  express  astronomically  a  given 
time  from  midnight  to  noon,  we  must  diminish  the 
number  of  the  common  day  by  a  unit,  and  increase  the 
number  of  the  hour  by  IS. 

3.  The  angle  contained  between  the  meridian  and 
a  declination  circle  passing  through  the  sun  or  any  one 
of  the  heavenly  bodies  is  called  th  v  Distance  of  the  body 
from  the  Meridian ,  or  the  Hour  angle  of  the  body.  The 
intercepted  arc  of  the  equator  is  the  measure  of  this  an¬ 
gle,  and  therefore  designates  the  distance  of  the  body 
from  the  meridian. 

4.  The  point  of  the  equator  which  is  on  the  meridian 
at  the  same  time  with  the  sun,  will,  by  the  diurnal  mo¬ 
tion,  be  15°  to  the  west  at  the  end  of  a  sidereal  hour. 
But  on  account  of  the  sun’s  increase  in  right  ascension, 
during  the  time,  its  distance  from  the  meridian  must 
be  less  than  15°.  The  sun  does  not  therefore  move 
from  the  meridian  at  the  rate  of  15°  in  a  sidereal  hour. 

But  as  the  interval  from  the  time  the  sun  is  on  the 
meridian  to  its  return  to  it  again  is  divided  into  24  so¬ 
lar  hours,  and  as  the  distance  is  360°,  if  we  suppose 
the  right  ascension  to  increase  uniformly  during  the 
day,  the  sun’s  diurnal  motion  from  the  meridian  must 
be  at  the  rate  of  15°  in  a  solar  hour. 

*  A.  M.  are  the  initials  of  Ante  Meridian ,  forenoon,  and  P.M.  of  Post 
Meridian,  afternoon. 


CHAPTER  VIII. 


73 

5.  Since  15°  of  the  sun’s  distance  from  the  meridian 
corresponds  to  1  solar  hour,  1°  must  correspond  to  4 
minutes  in  time,  1'  to  4  seconds,  and  1"  to  4  thirds. 
Hence  to  convert  the  sun’s  distance  from  the  meridian 
into  time,  if  we  multiply  the  distance  in  degrees,  mi¬ 
nutes  and  seconds,  by  4,  the  product  of  the  seconds  will 
be  thirds  of  time,  the  product  of  the  minutes  will  be 
seconds,  and  the  product  of  the  degrees  will  be  mi¬ 
nutes.  As  an  example  let  17°  21'  36"  be  converted 
into  time. 

17°  SI'  38" 

_ 4_ _ 

lh.  9m.  26  sec.  24  thirds. 

The  sun’s  distance  from  the  meridian  is  sometimes 
called  the  time  from  noon  in  degrees. 

6.  If  the  apparent  annual  motion  of  the  sun  was  in 
the  equator  and  was  uniform  at  the  rate  of  5 9'  8". 33 
from  the  mean  equinox,  in  the  interval  between  two  of 
its  consecutive  passages  over  the  meridian,  it  is  evident 
that  the  intervals  would  be  mean  solar  days  (7.9  and 
4).  But  the  motion  of  the  sun  is  not  in  the  equator,  and 
in  the  eeliptic  it  is  not  uniform.  There  are,  therefore, 
two  causes  of  inequality  in  mean  solar  days. 

7*  Supposing,  as  in  the  last  article,  the  sun  to  move 
uniformly  in  the  equator  with  its  mean  daily  motion  in 
longitude  from  the  mean  equinox,  the  time  when  it 
would,  in  that  case,  be  on  the  meridian,  is  called  Mean 
Noon.  And  time  reckoned  from  mean  noon  is  called 
Mean  Time. 

The  time  when  the  sun  is  really  on  the  meridian  is 
called  Apparent  Noon .  And  time  reckoned  from  ap¬ 
parent  noon  is  called  Apparent  Time. 

11 


74 


ASTRONOMY. 


The  difference  between  the  apparent  and  mean 
time  is  called  the  Equation  of  Time . 

8.  If  the  sun’s  mean  longitude  be  corrected  by  the 
equation  of  the  equinoxes  in  right  ascension,  The  differ¬ 
ence  between  the  true  right  ascension  and  the  corrected 
mean  longitude  will  be  the  equation  of  time  in  degrees. 

When  the  true  right  ascension  is  greater  than  the 
corrected  mean  longitude,  the  equation  of  time  must  be 
added  to  apparent  time  to  obtain  mean  time ,  and  when 
it  is  less,  the  equation  must  be  subtracted. 

But  on  the  contrary,  when  mean  time  is  to  be  re¬ 
duced  to  apparent  time,  the  equation  must  be  subtract¬ 
ed,  if  the  true  right  ascension  is  greater  than  the  cor¬ 
rected  mean  longitude,  and  must  be  added,  if  it  is  less . 

Let  VQ  Fig.  16,  be  the  equator,  VC  the  ecliptic,  V  the  ver¬ 
nal  equinox,  A  its  mean  place  in  the  ecliptic,  S  the  true  place  of 
the  sun,  and  AB  and  SD,  declination  circles.  Then  will  B  be 
the  mean  place  of  the  vernal  equinox  in  the  equator,  YA  be  the 
equation  of  the  equinoxes  in  longitude,  YB,  in  right  ascension, 
VS  the  sun’s  true  longitude,  and  VD  its  true  right  ascension. 
Also  let  BL  be  equal  to  the  sun’s  mean  longitude  from  the  mean 
equinox,  at  the  time  the  true  longitude  is  VS  or  true  right  ascen¬ 
sion  VD,  and  M  be  the  point  of  the  equator  which  is  on  the  me¬ 
ridian  at  that  time.  Then  (5  and  7), 

MD  =  apparent  time  in  degrees, 

ML  =  mean  time, 

and  DL  =  VD  —  VL  ==  equation  of  time. 

Now  (App.  49)  tan  VB  =  tan  VA  cos  V;  or  since  VA  and 
consequently  VB,  only  amounts  to  a  few  seconds  (7.54), 

VB  =  VA  cos  V  =  VA  cos  23°  28'  =  .917  VA. 

Hence  DL  =  VD  —  VL  =  VD  —  (LB  +  BV)  =  VD  — 
(LB  +  .917  VA). 

9.  If  the  equation  of  the  equinoxes  in  right  ascen- 


CHAPTER  VIII,  75 

sion  be  omitted,  the  error  in  the  equation  of  time  will 
seldom  exceed  1  second. 

Since  VA  when  greatest  is  only  18"  (7.54),  the  value  of  .917  VA 
can  not  exceed  16", 5,  or  1  second  6  thirds,  in  time. 

10.  The  equation  of  time  may  be  further  resolved 
into  its  component  parts.  If  E  be  the  equation  of  the 
centre,  P  the  amount  of  the  perturbations,  q  the  equation 
of  the  equinoxes  in  longitude,  and  R  the  reduction  to 
the  equator,  then, 

Equation  of  time  in  degrees  —  E  4-  R  +  P  4-  .083.^. 

Put 

M  =  sun’s  mean  longitude, 
and  a  =  obliquity  of  the  ecliptic; 
then  (11)  q  cos  =  equation  of  equinoxes  in  right  ascension. 

Hence, 

VL  =  RL  +  BV  =  M  4-  q  cos  <y, 

VS  =  M  +  E  +  P-t-<7  (7.54), 
and  VD  =  M+  E  +  P  +  ^-fR  (7.55). 

Therefore  DL  —  VD  —  VL  =  E-fP-f-R4-9  —  7  cos  u 
,  =  E  4.  P  4-  R  -f  q  ( 1  —  cos  «) 

=  E  +  P  -f  R  x  2  qsm  2  -1  a  (App.  8) 

=  E  4-  R  4-  P  -f  .083  q. 

11.  The  term  E  depends  on  the  sun’s  mean  anomaly, 
and  the  eccentricity  of  the  orbit  (7*30);  and  the  sun’s 
mean  anomaly  depends  on  its  mean  longitude  and  the 
longitude  of  the  apogee.  Therefore  the  term  E  de¬ 
pends  on  the  mean  longitude  of  the  sun,  longitude  of 
the  apogee,  and  the  eccentricity  of  the  orbit  The  lat¬ 
ter  two  change  but  little  in  a  year.  Hence  for  a  given 
year,  a  change  in  E,  depends  pricipally  on  the  mean 
longitude. 

With  a  given  obliquity  of  the  ecliptic,  the  term  R 
depends  on  the  sun’s  true  longitude  (7.55),  and  there- 


76  ASTRONOMY. 

fore  principally  on  the  mean  longitude  and  term  E. 

A  table  may  therefore  he  formed  containing  for  each 
degree  of  the  sun’s  mean  longitude,  the  sum  of  E  and 
R,  expressed  in  time. 

Such  a  table  will  only  he  true  for  the  time  for  which 
it  is  calculated.  But  the  greatest  error  only  amounts 
to  about  15  seconds  in  100  years.  Sometimes  a  column 
is  annexed,  containing  the  secular  variation  in  this  part 
of  the  equation  of  time. 

12.  The  term  P  must  consist  of  several  parts,  each 
depending  on  the  body  which  produces  the  perturba¬ 
tion.  The  greatest  value  of  P  is  about  45"  (7.53), 
which  is  3  seconds,  in  time.  The  greatest  value  of  the 
term  .83^,  is  about  TV  of  a  second  in  time.  In  calcu¬ 
lations  where  great  accuracy  is  not  required,  these 
terms  may  be  omitted. 

13.  When  E  and  R  are  equal  and  have  contrary 
signs,  their  sum  is  nothing.  Nearly  at  the  same  time, 
as  the  other  terms  only  amount  to  a  few  seconds,  the 
equation  of  time  must  also  be  nothing,  This  circum¬ 
stance  takes  place  four  times  in  the  year.  These 
times  are  about  the  15th  of  April,  15th  of  June,  1st  of 
September  and  24th  of  December. 

14.  As  the  motions  of  clocks  or  watches,  that  are 
well  made,  are  uniform,  or  nearly  so,  they  can  not  cor¬ 
respond  with  the  unequal  motion  of  the  sun.  They 
should  therefore,  for  the  common  purposes  of  life,  be 
regulated  to  mean  solar  time.  This  is  easily  done  by 
applying  the  equation  of  time,  to  the  observed  time  of 
the  sun’s  passage  over  the  meridian.* 

*  A  simple  method  of  drawing  a  meridian  line  that  will  be  sufficiently  ac. 
curate,  to  regulate  a  clock  for  the  common  purposes  of  society,  will  be  given 
in  the  next  chapter. 


CHAPTER  IX. 


77 


RIGHT  ASCENSION  OF  THE  MID-HEAVEN. 

15.  The  arc,  contained  in  the  order  of  the  signs,  be¬ 
tween  the  vernal  equinox  and  the  point  of  the  equator, 
which  is  on  the  meridian  at  any  time,  is  called  the 
Might  Ascension  of  the  Mid-heaven  at  that  time.  Thus 
VM  is  the  right  ascension  of  the  mid- heaven,  when 
the  point  M  is  on  the  meridian. 

16.  The  right  ascension  of  the  mid  heaven,  VM,  is 
equal  to  the  sum  of  VD  and  I)M;  that  is,  to  the  sum  of 
the  true  right  ascension  of  the  sun  and  the  apparant 
time  expressed  in  degrees . 

It  is  also  equal  to  the  sum  of  VB,  BL,  and  LM; 
that  is,  to  the  sum  of  the  equation  of  the  equinoxes  in 
right  ascension ,  mean  longitude  of  the  sun ,  and  the 
mean  time  expressed  in  degrees . 

In  either  case,  the  time  is  to  be  reckoned  from  noon 
to  noon;  and  if  the  sum  exceeds  360°,  its  excess  above 
360°,  is  the  right  ascension  of  the  mid- heaven. 

CHAPTER  IX. 

Circumstances  of  the  diurnal  motion. — Sun’s  Spots, 
and  rotation  on  its  axis. — Zodiacal  Light . 

1.  Let  HZNR,  Fig.  17,  represent  the  meridian  of  a 
place,  Z  the  zenith,  P  the  north  pole,  P'  the  south  pole, 
HR  the  horizon,  EQ  the  equator,  S  the  situation  of  a 
body  in  the  eastern  part  of  the  horizon,  and  PSD  a  de¬ 
clination  circle  passing  through  the  body. 

2.  If  PS,  the  polar  distance,  does  not  change,  the 
body  rising*  at  S  will  describe  the  arc  SB,  parallel  to 

*  It  has  been  noticed  that  the  rising  and  setting  of  the  heavenly  bodies 
aTe  affected  by  refraction  (3.20).  It  is  also  evident  that  for  the  sun,  moon, 
and  planets,  parallax  will  produce  some  effect.  But  in  this  and  the  following 


78 


ASTRONOMY. 


the  equator,  and  come  to  the  meridian  at  B.  In  descend¬ 
ing  from  the  meridian  to  the  horizon,  it  will  describe  a 
similar  and  equal  arc  in  the  western  hemisphere.  It 
will  therefore  be  the  same  length  of  time  in  descending 
from  the  meridian  to  the  horizon,  as  in  ascending  from 
the  horizon  to  the  meridian. 

3.  The  angle  SPZ,  or  its  measure  DE,  converted 
into  time,  expresses  the  interval  of  time  between  the 
rising  of  the  body  and  its  passage  over  the  meridian. 
This  interval  is  called  the  Semi-diurnal  Arc .  As  the 
body  is  12  hours  in  passing  from  A  to  B,  the  differ¬ 
ence  between  the  semi  diurnal  arc  and  12  hours,  ex¬ 
presses  the  time  in  which  the  body  ascends  to  the  hori¬ 
zon  from  the  meridian  below,  and  is  called  the  Semi- 
nocturnal  Arc. 

4.  When  PS  =  90°,  S  coincides  with  O,  and  the 
body  is  in  the  equator.  As  OPE  is  90°,  the  semi-diur¬ 
nal  arc  will  then  be  6  hours.  When  PS  is  less  than 
90°,  it  is  evident  the  angle  SPZ  will  be  more  than  90°, 
and  consequently  the  semi-diurnal  arc,  more  than  6 
hours.  When  PS  is  more  than  90°,  as  PS',  the  angle 
S'PZ  is  less  than  90°,  and  the  semi-diurnal  arc,  less 
than  6  hours. 

5.  When  PS,  the  distance  of  the  body  from  the  ele¬ 
vated  pole,  is  less  than  PH,  the  latitude  of  the  place, 
the  body  continues  above  the  horizon  and  does  not  set. 
When  the  distance  of  a  body  from  P',  the  depressed 
pole,  is  less  than  P'R,  which  is  equal  to  PH,  the  lati¬ 
tude  of  the  place,  the  body  continues  below  the  hori¬ 
zon  and  does  not  rise. 

6.  The  sun’s  polar  distance,  when  least,  is  about 
66°i.  Therefore,  at  a  place  whose  latitude  is  greater 

articles,  unless  when  the  contrary  is  mentioned,  these  effects  are  not  con¬ 
sidered. 


CHAPTER  IX. 


79 

than  66°|,  the  sun,  when  nearest  the  elevated  pole, 
will  revolve  above  the  horizon  without  setting,  and  will 
continue  to  do  so,  as  long  as  its  distance  from  the  ele¬ 
vated  pole  is  less  than  the  latitude.  At  the  opposite 
season  of  the  year,  when  the  sun  is  nearest  the  depress¬ 
ed  pole,  it  will  continue  a  like  period  of  time,  below 
the  horizon  without  rising. 

7.  At  either  of  the  poles  of  the  earth,  the  latitude  is 
90°;  and,  therefore,  the  sun  will  continue  above  the  hori¬ 
zon  all  the  time  it  is  on  the  same  side  of  the  equator 
with  the  elevated  pole,  which  is  about  half  the  year. 
During  the  other  half  of  the  year  it  will  be  below  the 
horizon. 

8.  At  the  equator,  the  latitude  is  nothing,  the  axis 
PP'  coincides  with  HR,  and  the  angle  SPZ  becomes 
SHZ  =  90°.  The  semi-diurnal  arc  is  therefore  6 
hours,  whatever  be  the  distance  of  the  body  from  the 
pole. 

9.  It  follows  from  the  preceding  articles,  that  at  the 
equator,  the  days*  are  always  12  hours  long;  from  the 
equator  to  86°|  latitude,  the  longest  day  varies  from 
12  to  24  hours;  and  from  thence  to  the  pole  it  varies 
from  24  hours  to  6  months. 

10.  The  difference  in  the  warmth  diffused  by  the 
sun  to  different  parts  of  the  earth,  or  to  the  same  part 
at  different  seasons  of  the  year,  depends  principally  on 
its  continuance  above  the  horizon,  and  on  its  meridian 
altitude.  The  change  in  the  sun’s  distance  from  the  earth 
must  produce  some  effect;  but  on  account  of  the  small  de¬ 
gree  of  eccentricity  of  the  earth’s  orbit,  the  change  in 
distance  is  only  a  small  part  of  the  whole  distance,  and 
consequently  the  difference  in  warmth  depending  on 

*  The  term  day,  here,  implies  the  time  of  the  sun’s  continuance  above 
the  horizon. 


80 


ASTRONOMY. 


this  cause,  will  be  inconsiderable.  Indeed  the  sun  is  in 
perigee,  or  nearest  the  earth,  about  the  first  of  January; 
which,  in  northern  latitudes,  is  the  time  of  our  coldest 
weather. 

11.  Within  the  limits  of  the  torrid  zone  (0.10), 
when  the  sun  is  on  the  meridian,  the  direction  of  its 
rays  is  always  nearly,  and  sometimes  quite,  perpen¬ 
dicular  to  the  surface  of  the  earth.  The  heat  is  there¬ 
fore  very  great. 

On  the  contrary,  within  the  frigid  zones,  the  sun 
never  rising  far  above  the  horizon,  the  direction  of  its 
rays  is  very  oblique,  and  consequently  it  produces  but 
little  warmth. 

Within  the  temperate  zones,  the  obliquity  in  the  di¬ 
rection  of  the  sun’s  rays,  when  it  is  on  the  meridian,  be¬ 
ing  never,  either  very  great  or  very  small,  a  medium 
temperature  is  produced. 

12 .  At  any  place  in  the  temperate  zones,  the  differ¬ 
ence  between  the  greatest  and  least  meridian  altitudes 
of  the  sun,  is  evidently  equal  to  twice  the  greatest  de¬ 
clination,  that  is,  to  nearly  1/°.  And  as  the  days  are 
longest  when  the  meridian  altitude  is  greatest,  and 
shortest  when  it  is  least,  the  difference  in  temperature, 
between  the  opposite  seasons  of  the  year,  in  which 
these  circumstances  have  place,  must  be  considerable. 

13.  In  the  north  temperate  zone,  Spring ,  Summer , 
Autumn,  and  Winter ,  the  four  seasons,  into  which 
the  year  is  divided,  are  considered  as  respectively 
commencing  at  the  times  of  the  Vernal  Equinox , 
Summer  Solstice ,  Autumnal  Equinox ,  and  Winter 
Solstice. 

14.  Since  at  the  terrestrial  equator,  the  poles  are  in 
the  horizon,  the  celestial  equator,  and  circles  parallel 
to  it,  must  cut  the  horizon  at  right  angles.  This  posi- 


CHAPTER  IX. 


81 


tion  of  the  sphere  is  called  a  Right  Sphere.  At  the 
poles,  the  equator  coincides  with  the  horizon;  and 
hence,  circles  parallel  to  it,  are  parallel  to  the  horizon. 
This  is  called  a  Parallel  Sphere.  At  any  other  place 
on  the  earth,  the  equator  and  its  parallels  cut  the  hori¬ 
zon  obliquely;  and  such  a  position  of  the  sphere  is  call¬ 
ed  an  Oblique  Sphere. 

15.  A  circle  parallel  to  the  horizon  is  called  ail  A l- 
macantar. 

10.  The  arc  of  the  equator,  contained  in  the  order 
of  the  signs,  between  the  vernal  equinox  and  the  point 
of  the  equator,  which  is  in  the  horizon  at  the  same  time 
with  any  body,  is  called  the  Oblique  Ascension  of  the 
body.  Thus,  if  Y  be  the  place  of  the  vernal  equinox, 
YO  is  the  oblique  ascension  of  the  body  at  S. 

17.  The  arc,  which  is  the  difference  between  the 
right  ascension  of  a  body  and  the  oblique  ascension,  is 
called  the  Ascensional  Difference.  Thus,  OD  is  the 
ascensional  difference  of  the  body  at  S. 

18.  Given  the  latitude  of  the  place  and  the  Bull’s  declination ,  to 
find  the  time  of  its  using  or  setting . 

In  the  triangle  ZPS  we  have  (App.  34), 

cos  ZS  =  cos  PZ  cos  PS  4-  sin  PZ  sin  PS  cos  SPZ, 

or  because  ZS  =  90°, 

0  =  cos  PZ  cos  PS  4-  sin  PZ  sin  PS  cos  SPZ. 

Hence  cos  SPZ  =  —  cos  Pf  =  —  cot  PZ  cot  PS  = 

sin  PZ  sin  PS 

—  tan  PH  cot  PS, 

or  cos  semi-diurnal  arc  —  —  tan  latitude  x  cot  polar  dis¬ 
tance. 

Another  formula  which  is  frequently  used,  may  be  derived  from 
the  preceding. 

Because,  cos  SPZ  =  cosDE  =  cos  (90°  4-  OD)  =  — sin  OD, 

12 


82 


ASTRONOMY. 


we  have,  sin  OD  =  tan  PH  cot  PS  =  tan  PH  tan  DS, 

or  sin  Ascensional  dijj.  ...  tan  latitude  x  tan  declination . 

Now  the  angle  OPE  or  its  measure  OE,  being  equal  to  90°  or 
6  hours,  it  is  evident  that  when  the  latitude  and  declination  are 
both  of  the  same  name,  that  is,  both,  north  or  both  south,  the  as¬ 
censional  difference  converted  into  time  and  added  to  6  hours  will 
give  the  semi-diurnal  arc. 

When  the  latitude  and  declination  are  of  different  names,  the 
ascensional  difference  in  time,  subtracted  from  6  hours,  gives  the 
semi-diurnal  arc. 

The  semi-diurnal  arc  expresses  the  time  of  sunset,  and  sub¬ 
tracted  from  12  hours,  gives  the  time  of  sunrise. 

19.  Given  the  latitude  of  the  place  and  the  sun's  declination  to 
find  its  azimuth  at  rising  or  setting. 

In  the  triangle  ZPS  we  have  (App.  34) 

cos  PS  ==  cos  PZ  cos  ZS  -f  sin  PZ  sin  ZS  cos  PZS, 

or  because  ZS  =  90°, 

cos  PS  =  sin  PZ  cos  PZS. 

Hence  cos  PZS  =  ..cosPS  =  cos  PS  , 
sin  PZ  cos  PH’ 

or  cos  azimuth  = 

cos  latitude 


20.  Given  the  latitude  of  the  place ,  the  sun's  altitude *  and  de¬ 
clination,  to  find  the  time  of  day. 

Let  S,  Fig.  18,  be  the  situation  of  the  sun;  and  put, 

L  =  HP  =  the  latitude, 

D  =  PS  =  90°  ±  sun’s  declination, 

H  =  SE  =  the  sun’s  altitude, 

P  =  ZPS  =  the  hour  angle. 

Then  in  the  triangle  ZPS,  we  have  (App.  34), 
cos  ZS  —  cos  PS  cos  ZP 


cos  ZPS  = 


sin  PS  sin  ZP 


*  The  sun’s  altitude  may  be  obtained  with  considerable  accuracy  by  means 
of  a  Sextant  of  Refection  and  an  Artificial  Horizon.  For  a  description  of  these 
and  the  manner  of  using  them,  the  student  is  referred  to  Boxvditch's  Navi¬ 
gation. 


CHAPTER  IX. 


83 


or  cos  P 


cos  (90°  —  H)  —  cos  D  cos  (90°-; — J-.) 

sin  D  sill  (90°  —  L) 


sin  H  —  cos  D  sin  L 

«=  - - - 

sin  D  cos  L 

But  (App.  8),  cos  P  =  1  —  2  sin  2  ^  P, 

and  (App.  13),  cos  D  sin  L  =  sin  (D  +  L)  —  sin  D  cos  L. 

Hence 


2  sin 


2  sin2  a  p  _.  sin  H  —  (sin  D  -f  L)  -f  sin  P  cos  L 

sin  D  cos  L 

sin  H  — sin  (D  +  L)  1 

—  - ; — - - - t-  1  •> 

sin  D  cos  L 

,  p  sin  (D  -f  L)  —  sin 

“a  1  =  - - - - - - - 


H 


sin  D  cos  L 


2  cos \  (D  +  L  +  H)sinl(D  +  L  — H) 


sin  D  cos  L 


(App.  21) 


2  cos 

V  2 


)  Si“( 


D  +  L  +  H 


-H) 


sin  D  cos  L 


cos 


D  +  L  4.  H\ 


sin 


D  +  L  +  H 


H 


sin  i  P  =  v/ 


sin  D  cos  L 


To  determine  the  time  accurately,  the  observation  of  the  sun’s 
altitude  should  not  be  made  when  the  sun  is  near  the  meridian,  as 
its  altitude  then  changes  but  slowly.  Neither  should  it  be  made 
when  the  sun  is  very  near  the  horizon,  as  the  correction  for  re¬ 
fraction  can  not  then  be  depended  on  with  certainty.  In  general 
the  best  time  is  three  or  four  hours  before  or  after  noon. 

It  is  the  true  altitude  of  the  sun’s  centre  that  is  to  be  used  in 
the  calculation.  Hence  the  altitude  of  the  lower  or  upper  limb, 
as  obtained  by  observation  must  be  corrected  for  refraction,  pa¬ 
rallax,  and  semi-diameter. 


21.  If  the  sun’s  declination  did  not  change,  it  is  evi¬ 
dent  that  it  would  have  equal  altitudes  at  equal  times 
before  and  after  apparent  noon.  Hence  if  an  observa¬ 
tion  of  the  sun’s  altitude  was  taken  in  the  forenoon,  and 
the  time  observed  by  a  good  clock  or  watch,  and  if  in 


84 


ASTRONOMY. 


the  afternoon  the  time  was  also  observed  when  the  sun 
had  obtained  the  same  altitude;  half  the  interval  added 
to  the  time  of  the  first  observation,  would  give  the  time 
shown  by  the  clock  or  watch  when  the  sun  was  on  the 
meridian.  The  deviation  from  12  o’clock  would  be 
the  error  of  the  clock  with  respect  to  apparent  time. 

But  in  consequence  of  the  sun’s  change  in  declina¬ 
tion  during  the  interval  between  the  observations,  it  is 
necessary,  in  order  to  render  this  method  accurate,  to 
apply  a  correction  to  the  time  thus  obtained.  This  cor¬ 
rection  is  called  the  Equation  of  Equal  Altitudes, 
Tables  have  been  calculated,  from  which  the  equation 
is  easily  obtained.  With  these  tables,  the  method  of 
obtaining  the  error  of  the  clock  by  equal  altitudes  of 
the  sun,  is  simple,  and  it  is  also  very  accurate. 


22.  Given  the  latitude  of  the  place ,  the  sun’s  altitude  and  decli¬ 
nation^  to  find  its  azimuth. 

Let  Z  =  PZS  ==  the  sun’s  azimuth;  then  (App.  34), 
cos  PS  —  cos  PZ  cos  ZS 


cos  PZS  == 


or  cos  Z  == 


sin  PZ  sin  ZS 
cos  D  —  sin  L  sin  H 


cos  L  cos  H 


But  (App.  9), 
cos  Z  2  cos2|-  Z 


L 


and  (App.  14),  sin  L  sin  II  =  cos  L  cos  H  —  cos  (L  -f-  H) , 
Hence, 

2  C0g2 1  ^ _ i  _  cos  D  4-  cos  (L  -f  H)  —  cos  L  cos  II 

cos  L  cos  H 

_  cos  D  -f  cos  (L  4-  H) _ j 

cos  L  cos  H 

2  cos2--  Z  =  cos  (L  +  H)  +  cos  D 
cos  L  cos  H 

_  2  cos  -J  (L  +  H  4-  D)  cos-i  (L  +  H  —  D) 
cos  L  cos  H 


(App.  22) 


CHAPTER  IX. 


2  COS 


COS 

cos  \  Z  =  s/  — 

23.  If  the  sun’s  declination  did  not  change,  it  is  evi¬ 
dent  that  equal  azimuths  would  correspond  with  equal 
altitudes.  As  the  change  in  declination  is  but  little, 
for  a  few  hours,  particularly  near  the  time  of  the  sol¬ 
stices,  the  azimuths  corresponding  to  equal  altitudes, 
must  be  nearly  equal.  This  circumstance  furnishes  a 
simple  method  of  drawing  a  meridian  line  that  will  an¬ 
swer  for  determining  the  time  of  apparent  noon,  when 
great  accuracy  is  not  required.  To  do  this,  describe  a 
number  of  concentric  circles  or  arcs  of  circles  on  a 
smooth  board.  At  the  common  centre  of  these  arcs, 
fix  a  piece  of  thick,  straight  wire,  and  make  it  exactly 
perpendicular  to  the  surface  of  the  board.  By  the  aid 
of  a  spirit  level  or  even  of  a  common  plumb  line  level, 
fix  the  board  so  that  its  upper  surface  may  be  horizon¬ 
tal. 

On  a  clear  day,  observe,  during  the  forenoon,  when 
the  extremity  of  the  shadow  cast  by  the  wire,  exactly 
coincides  with  one  of  the  arcs,  and  mark  the  place.  In 
the  afternoon,  observe  when  the  extremity  of  the  sha¬ 
dow  coincides  with  the  same  arc.  A  straight  line 
drawn  from  the  place  of  the  wire,  through  the  middle 
point  of  the  arc  contained  between  the  marks  will  be  a 
meridian  line.  When  the  shadow  of  the  wire  coincides 
with  this  line,  it  is  apparent  noon. 

Greater  accuracy  will  be  obtained  by  extending  the 
observation  to  several  of  the  concentric  arcs,  and  if 


So 


P  +_L_+_H)  cos  ( 


D  +  L  +  H 

2 


°) 


cos  L  cos  H 

JD±L±H)  cqs  |DrL  +  H_D| 
cos  L  cos  H 


ASTRONOMY. 


8(5 

they  do  not  give  the  same  line,  taking  for  the  meridian 
line,  a  mean  between  them. 


24.  To  find  the  time  of  the  sun’s  apparent  rising  or  setting. 

At  the  time  of  the  apparent  rising  or  setting  of  the  sun,  the 
zenith  distance  ZS  =  90°  -f-  refraction  —  parallax.  Let  R  = 
refraction  — parallax;  then  ZS  =  90°  -f  R,  and  by  an  investi¬ 
gation  nearly  similar  to  that  in  article  20th,  we  have, 

D  +  L  +_Rj  cos  +  L.+.R  _ 
sin  D  cos  L 

As  it  is  not  important  to  know  the  precise  time  of  the  rising  or 
setting  of  the  heavenly  bodies,  it  is  usual  to  omit  the  effects  of 
refraction  and  parallax  and  to  consider  the  bodies  as  rising  or 
setting  when  they  are  really  in  the  horizon. 

25.  To  find  the  time  of  the  beginning  or  end  of  twilight. 

Twilight  commences  or  ends  when  the  sun  is  about  18°  below 

the  horizon.  Therefore  the  zenith  distance  ZS  =  90°  +  18°; 
and  by  substituting  18°  instead  of  Rin  the  formula  in  the  last  ar¬ 
ticle,  we  have, 


sm 


siniP=  v/. 


>i»  (D4-_L±J£)co,(D+_L  +  jy_  l8.) 

2  P  =  - ; - fr - j - 

sin  D  cos  L 


If  the  time  of  the  commencement  of  twilight  be  subtracted 
from  the  time  of  sunrise,  the  remainder  will  be  the  duration  of 
twilight. 


26.  The  duration  of  twilight  at  a  given  place, 
changes  with  the  declination  of  the  sun.  In  northern 
latitudes,  it  is  longest  when  the  sun  has  its  greatest 
north  declination;  and  shortest  when  the  declination  is 
a  few  degrees  south.  It  is  not  designed  to  enter  into 
an  explanation  of  the  different  circumstances,  relative 
to  the  duration  of  twilight,  as  they  are  of  but  little  prac¬ 
tical  utility.  But  the  determination  of  the  time  of 


CHAPTER  IX. 


87 

shortest  twilight,  being  a  problem  that  has  claimed  con- 
siderable  attention,  may  be  introduced. 

27.  Given  the  latitude  of  the  place ,  to  determine  the  duration  of 
the  shortest  tioilight  and  the  sun's  declination  at  the  time . 

In  the  solution  of  this  problem,  twilight  is  supposed  to  com¬ 
mence  when  the  sun  is  at  a  given  distance  below  the  horizon,  the 
sun  to  rise  when  it  is  really  in  the  horizon,  and  its  distance  from 
the  pole  to  remain  constant,  during  the  continuance  of  twilight. 
The  sun’s  distance  below  the  horizon  when  twilight  commences 
is  generally  assumed  to  be  18°. 

Put  2a  =  18°,  and  let  S  Fig .  19,  be  the  situation  of  the  sun 
when  twilight  commences,  and  S'  its  situation  in  the  horizon. 
Then  PS  =  PS',  ZS'  =  90°,  ZS  =  ZD  -f  DS  =  90°  +  2  a, 
and  the  angle  SPS',  converted  into  time,  expresses  the  duration 
of  twilight.  Let  PCS  be  a  spherical  triangle  having  the  sides 
respectively  equal  to  the  sides  of  the  triangle  ZPS',  that  is,  PS  = 
PS',  PC  =  PZ  and  CS  ==  ZS'  =  90°.  Also  let  ZC  be  the  arc 
of  a  great  circle  joining  Z  and  C.  Then  the  angle  CPS  is  equal 
to  ZPS',  and  consequently  ZPC  =  SPS'.  Hence  when  the 
angle  ZPC  is  the  least  possible,  the  twilight  will  be  shortest. 

Now  in  the  triangle  ZPC,  the  two  sides  ZP  and  PC  are  con¬ 
stant;  and  therefore  the  angle  ZPC  will  be  least  when  the  side 
ZC  is  least.  But  as  the  two  sides  CS  and  ZS  of  the  triangle 
ZSC  are  constant,  the  side  ZC  will  be  least  when  the  angle 
ZSC  =  0;  that  is  when  the  sides  ZC  and  CS  coincide  with  ZS. 
Hence  when  the  twilight  is  shortest,  the  angle  PS'Z  =  PSC  = 
PSZ. 

We  have,  Fig.  20,  ZC  =  ZD  —  CD  =  CS  —  CD  =  DS 
==  2  a.  And  because  PZ  =  PC,  if  PE  bisect  the  angle  ZPC, 
it  will  also  bisect  the  base  ZC,  and  be  perpendicular  to  it. 
Hence, 

sin  ZPE  —  sin  —  s‘n  a  _  sin  9° 

sin  PZ  cos  PH  cos  latitude 

Twice  the  angle  ZPE  converted  into  time,  gives  the  duration 
of  shortest  twilight. 


88 


ASTRONOMY. 


Since  the  two  right  angled  triangles  ZPE  and  SPE  have  the 
same  perpendicular  PE,  we  have  (App.  45.), 
cos  ZE  :  cos  ES  :  :  cos  PZ  :  cos  PS, 
cos  a  :  cos  (90°  -f  a)  :  :  sin  latitude  :  sin  declination , 

or  cos  a  :  —  sin  a  :  :  sin  latitude  :  sin  declination. 

Hence  sin  declination  =  —  — n— ?  sin  latitude  =  —  tan  a  sin 

cos  a 

latitude. 

The  value  of  sin  declin.  being  negative,  shows  that  the  decli¬ 
nation  is  of  a  different  name  from  the  latitude  of  the  place. 
Hence  in  northern  latitudes  the  declination  is  south. 

The  times  of  the  year,  at  which  the  shortest  twilight  has  place 
may  be  ascertained  by  observing  in  a  Nautical  Almanac*,  the 
days  on  which  the  sun  has  the  declination,  found  by  the  above 
formula. 

28.  The  ancients  gave  considerable  attention  to  the 
rising  or  setting  of  a  star  or  planet  under  the  circum¬ 
stances  noticed  in  the  following  definitions.  But  it  is 
not  now  considered  of  much  importance. 

29.  The  Cosmical  rising  or  setting  of  a  star,  is  when 
it  rises  or  sets  at  sunrise. 

The  Jlchronical  rising  or  setting  of  a  star,  is  when  it 
rises  or  sets  at  sunset. 

The  Heliacal  rising  of  a  star,  is  when  it  rises  so 
long  before  sunrise  as  just  to  become  visible  above  the 
horizon  and  then  immediately  to  disappear  in  conse¬ 
quence  of  the  increasing  light  from  the  sun;  and  its 
Heliacal  setting,  is  when  it  sets  so  long  after  the  sun 
as  just  to  become  visible  before  it  descends  belo^v  the 
horizon. 

*  The  Nautical  Almanac  is  a  work  published  annually  in  England,  and  at 
present  republished  in  New  York,  and  may  generally  be  obtained  one  or 
two  years  previous  to  the  year  for  which  it  is  calculated. 


CHAPTER  IX. 


89 


sun’s  spots,  and  rotation  on  its  axis. 

30.  The  sun  presents  to  us  the  appearance  of  a  lu¬ 
minous,  circular  disc.  But  it  does  not  from  thence  fol¬ 
low  that  its  surface  is  really  flat;  for  all  globular  bo¬ 
dies  when  viewed  at  a  great  distance  have  this  ap¬ 
pearance.  Observations  with  a  telescope  show  that 
the  sun  has  a  rotatory  motion.  And  it  is  only  a  globu¬ 
lar  body,  that  in  presenting  all  its  sides  to  us,  would 
always  appear  under  the  form  of  a  circle. 

31.  When  the  sun  is  viewed  with  a  telescope,  dark 
Spots  of  an  irregular  form  are  often  seen  on  its  disc; 
and  continued  or  repeated  observations  show  that  they 
have  a  motion  from  east  to  west.  Their  number,  po¬ 
sitions  and  magnitudes  are  extremely  variable.  Fre¬ 
quently,  several  may  be  seen  at  once;  and  at  some  pe¬ 
riods,  for  a  year  or  more  there  are  none  visible.  Their 
magnitude  is  sometimes  such  as  to  render  them  visible 
to  the  naked  eye,  when  in  consequence  of  a  smoky  or 
thick  atmosphere,  the  sun  can  be  thus  viewed  without 
injury.  Each  spot  is  usually  surrounded  with  a  pe¬ 
numbra,  beyond  which  is  a  border  of  light  more  bril¬ 
liant  than  the  rest  of  the  sun’s  disc.  Sometimes  a  spot 
becomes  visible  on  the  eastern  limb  of  the  sun,  tra¬ 
verses  the  disc  in  about  14  days,  disappears  in  the 
west,  and  after  a  like  interval  reappears  in  the  east. 
But  it  is  not  often  that  this  happens,  as  the  spots  fre¬ 
quently  dissolve  and  perish  before  they  arrive  at  the 
western  side;  or  having  disappeared  on  that  limb,  do 
not  reappear.  The  nature  of  the  solar  spots  and  the 
causes  of  these  changes  are  to  us  unknown. 

3&.  When  a  spot  remains  so  long  permanent,  as  to 
be  seen  twice  in  the  same  position  on  the  sun’s  disc, 
the  interval  is  found  to  be  about  27  days.  But  this 

13 


ga 


ASTRONOMY. 


interval  does  not  express  the  real  period  of  the  sun’s 
rotation  on  its  axis.  For  during  this  time  the  sun,  by 
its  apparent  annual  motion,  has  advanced  nearly  a  sign 
forward  in  the  ecliptic.  The  spot  has,  therefore,  made 
that  much  more  than  a  complete  revolution,  before  it 
appears  to  a  spectator  on  the  earth,  to  be  in  the  same 
position. 

33.  The  apparent  position  of  a  spot  with  respect  to 
the  sun’s  centre  may  be  obtained  by  observing,  when 
the  sun  is  on  the  meridian,  the  right  ascensions  and  de¬ 
clinations,  both  of  the  spot  and  centre.  From  three  or 
more  observations  of  this  kind,  the  time  of  the  sun’s 
rotation  and  the  situation  of  its  equator  with  respect  to 
the  ecliptic,  may  be  ascertained.  The  student  who 
wishes  a  complete  investigation  of  these,  may  be  rc- 
fered  to  the  astronomy  of  Delambre  or  Biot.  The  fol¬ 
lowing  results  have  been  obtained  by  Delambre. 

The  time  of  the  sun’s  rotation  on  its  axis  25  d.  0  h. 
17  m.  Inclination  of  the  sun’s  equator  to  the  ecliptic 
7°  19'.  The  north  pole  of  the  sun  is  directed  towards 
a  point  in  the  ecliptic,  the  longitude  of  which  is  11s. 
£0°.  Some  astronomers  make  the  time  of  the  sun’s  ro 
tation  on  its  axis  25  d.  12  h. 

34.  It  is  also  found  that  the  moon  and  such  of  the 
planets  as  admit  of  sufficiently  nice  observations  to  de¬ 
termine  the  fact,  have  motions  on  their  axes.  This 
forms  a  strong  analogical  proof  in  favour  of  the  earth’s 
diurnal  motion. 


ZODIACAL  LIGHT. 

3 5.  A  luminous  appearance  is  sometimes  seen  after 
sunset  or  before  sunrise,  in  the  form  of  a  cone  or  pyra¬ 
mid,  with  its  base  at  that  part  of  the  horizon  which  the 
sun  has  just  left  or  at  which  it  is  about  to  appear,  and 


CHAPTER  X. 


91 


having  its  axis  in  the  same  direction  as  the  plane  of 
the  sun’s  equator.  This  phenomenon  is  called  the 
Zodiacal  Light.  From  the  circumstance  of  its  direc¬ 
tion  always  corresponding  with  the  sun’s  equator,  it 
seems  to  have  some  connection  with  the  sun’s  rotation. 

36.  The  angle  which  the  plane  of  the  sun’s  equator 
makes  with  the  horizon  of  a  given  place,  at  the  time  of 
sunset  or  sunrise,  is  different  for  different  positions  of 
the  sun  in  the  ecliptic.  In  our  northern  climates  the 
greatest  inclination,  at  the  time  of  sunset,  has  place 
about  the  1st  of  March;  and  at  this  season  of  the  year, 
the  zodiacal  light  is  generally  most  distinct.  At  other 
seasons  when  the  inclination  is  less,  the  vapours  near 
the  horizon  conceal  it  from  our  view.  On  account  of 
the  different  states  of  the  air,  at  the  season  most  fa¬ 
vourable  to  its  appearance,  it  is  much  more  distinct  in 
some  years,  than  in  others. 

The  extent  of  the  zodiacal  light  is  various,  being 
sometimes  more  than  100°,  and  sometimes  not  more 
than  40°  or  50°. 

CHAPTER  X. 

Of  the  Moon . 

1.  T.  re  moon,  next  to  the  sun,  is  the  most  conspicu¬ 
ous  of  the  heavenly  bodies,  and  is  particularly  dis¬ 
tinguished  by  the  periodical  changes,  to  which  its  figure 
and  light  are  subject.  The  different  appearances  which 
it  presents,  are  called  the  Phases  of  the  moon. 

2.  By  repeatedly  observing  the  moon,  when  on  the 
meridian,  it  is  found  that  it  has  a  motion  among  the 
fixed  stars,  from  west  to  east,  and  that  it  comes  to  the 
meridian  about  50  minutes  later  oneach  succeeding  day. 
This  motion  is  not  uniform,  hut  at  a  mean,  it  is  13°  10' 


92 


ASTRONOMY. 


35"  in  24  mean  solar  hours.  It  is  also  found,  that  the 
moon  is  sometimes  on  the  north,  and  sometimes  on  the 
south  side  of  the  ecliptic,  continuing  about  as  long  on 
one  side  as  on  the  other;  and  that  its  orbit  nearly  coin¬ 
cides  with  the  plane  of  a  great  circle,  which  intersects 
the  ecliptic  in  an  angle  of  about  5°. 

3.  The  points  in  which  the  moon’s  orbit  cuts  the 
plane  of  the  ecliptic,  are  called  the  moon’s  Nodes. 
That  node  in  which  the  moon  is,  when  passing  from 
the  south  to  the  north  side,  is  called  the  Ascending 
Node .  The  other,  in  which  it  is,  when  passing  from 
the  north  to  the  south  side,  is  called  the  Descending 
Node .  The  nodes  are  distinguished  by  the  following 
characters. 

Ascending  node,  &. 

Descending  node,  £5. 

4.  At  periods  of  about  a  month  each,  the  moon  en¬ 
tirely  disappears,  and  continues  invisible  during  two  or 
three  days.  About  the  middle  of  this  time,  the  longi¬ 
tudes  of  the  sun  and  moon  are  equal.  It  is  then  said  to 
be  *  ew  Moon. 

5.  When  the  moon  again  becomes  visible,  it  is  seen 
soon  after  sunset,  a  little  above  the  western  part  of  the 
horizon,  under  the  form  of  a  circular  segment,  the  ex¬ 
terior  boundary  being  a  semicircle,  and  the  interior,  a 
semi-ellipse,  having  for  its  greater  axis,  the  diameter 
of  the  semicircle.  This  phase  of  the  moon  is  called  a 
Crescent ,  and  is  represented  in  Fig.  21.  The  points 
A  and  B  are  called  the  Cusps ,  or  Horns . 

6.  The  convex  part  of  the  crescent  is  turned  towards 
the  sun;  and  if  a  great  circle  bisect,  at  right  angles,  the 
line  AB,  which  joins  the  cusps,  it  will  pass  through 
the  sun. 


CHAPTER  X. 


93 


7.  From  clay  to  day  the  luminous  segment  increases 
in  breadth,  the  interior  boundary  becomes  less  concave, 
and  the  moon  advances  to  the  east  of  the  sun,  till  in  a 
little  more  than  seven  days  from  the  time  of  new  moon, 
the  difference  of  their  longitudes*  is  90°.  This  situa¬ 
tion  of  the  moon  with  respect  to  the  sun,  is  called  the 
First  Quarter.  Nearly  at  the  same  time  the  moon  ap¬ 
pears  as  a  semicircle,  the  right  line,  joining  the  cusps, 
becoming  the  boundary  on  the  side  opposite  the  sun. 
The  moon  is  then  said  to  be  Dichotomized ,  or  to  be  a 
Half  Moon. 

8.  After  this,  the  side  opposite  the  sun  becomes  con¬ 
vex,  and  the  convexity,  as  well  as  the  breadth  of  the 
segment,  increases  till  the  longitudes  of  the  sun  and 
moon  differ  180°,  which  is  in  about  fifteen  days  from 
new'  moon.  At  this  time  the  moon  appears  nearly  as  a 
complete  circle.  It  is  then  said  to  be  Full  Moon. 

A  phase  of  the  moon,  between  the  first  quarter  and 
fall  moon,  is  represented  in  Fig.  22.  When  the 
moon  appears  under  this  shape,  it  is  said  to  be  Gib¬ 
bous. 

9.  After  full  moon,  the  western  side  of  the  moon  be¬ 
comes  elliptical,  and  the  convexity  and  breadth  de¬ 
crease.  In  about  twenty  two  days  from  the  time  of 
new7  moon,  the  longitudes  of  the  sun  and  moon  differ 
270°.  It  is  then  said  to  be  Last  Quarter.  About  this 
time  the  moon  is  again  dichotomized. 

After  this,  the  western  side  of  the  moon  becomes 
concave,  and  the  breadth  of  the  segment  continues  to 
decrease  till  the  moon  again  becomes  invisible,  a  day 
or  two  before  new7  moon. 

*  By  the  difference  of  their  longitudes,  is  meant  the  excess  of  the 
moon’s  longitude  above  that  of  the  sun,  the  former  being  increased  by  360°, 
when  it  is  less  than  the  latter. 


ASTRONOMY. 


10.  The  interval  from  new  moon  to  new  moon,  or 
from  full  moon  to  full  moon,  is  called  a  Lunation ,  or 
Lunar  Month .  Its  mean  length  is  29  d.  12  h.  44  m. 
3  sec. 

11.  Any  two  of  the  heavenly  bodies  are  said  to  be 
in  Conjunction ,  when  their  longitudes  are  the  same; 
and  to  be  in  Opposition,  when  their  longitudes  differ 
180°.  The  points  in  the  orbit,  at  which  the  moon  is, 
when  in  conjunction  or  opposition,  are  called  the 
Syzigies ;  those  at  which  it  is,  when  its  longitude  ex¬ 
ceeds  the  sun’s  by  90°  or  270°,  are  called  the  Quadra¬ 
tures9,  and  the  middle  points,  between  the  syzigies  and 
quadratures,  are  called  the  Octants .  Some  of  these 
are  designated  by  characters,  as  follows: 

Conjunction,  <5, 

Opposition,  <?, 

Quadrature,  a  • 

12.  The  time  between  two  consecutive  conjunctions 
or  oppositions,  of  a  body  with  the  sun,  is  called  a  Sy- 
nodic  Revolution  of  that  body.* 

13.  It  follows  from  the  preceding  articles,  that  the  new 
moon  has  place,  when  the  moon  is  in  conjunction  with 
the  sun;  the  full  moon,  when  it  is  in  opposition;  and 
that  the  first  and  last  quarters  have  place  when  it  is  in 
quadratures.  Also,  that  the  synodic  revolution  of  the 
moon  is  the  same  with  a  lunation  or  lunar  month. 

14.  For  a  few  days  before  and  after  new  moon,  we 

*  As  the  distances  of  the  planets,  Mercury  and  Venus,  from  the  sun,  are 
each  less  than  that  of  the  earth  (7.28),  they  can  never  be  in  opposition  to 
the  sun.  But  they  may  be  in  conjunction,  either  by  being  between  the  sun 
and  earth,  or  by  being  on  the  opposite  side  of  the  sun.  The  former  is  called 
the  Inferior,  and  the  latter  the  Superior  conjunction.  For  either  of  these  pla¬ 
nets,  a  Synodic  Revolution  is  the  interval  between  two  consecutive  conjunc¬ 
tions  of  the  same  kind. 


CHAPTER  X. 


95 


can  discern  the  entire  disc,  the  obscure  part  appearing 
with  a  faint  dusky  light.  Near  the  time  of  full  moon, 
this  part  is  entirely  invisible. 

15.  When  the  moon  is  viewed  with  a  telescope,  the 
line  separating  the  light  part  from  the  dark,  is  seen  to 
be  very  irregular  and  serrated;  and  its  form  varies  even 
daring  the  time  of  observation.  Bright  spots  are  fre¬ 
quently  seen  on  the  dark  part,  near  the  line  of  separa¬ 
tion.  The  light  of  these  spreads,  till  it  becomes  united 
to  the  rest.  The  whole  enlightened  surface  also  ap¬ 
pears  diversified,  with  spots  of  different  shapes  and  dif¬ 
ferent  degrees  of  brightness.  These  spots  always  re¬ 
tain  the  same  position  with  respect  to  each  other,  and 
occupy  nearly  the  same  place  on  the  disc.  It  therefore 
follows,  that  nearly  the  same  face  of  the  moon  is  always 
turned  towards  the  earth. 

16.  A  consideration  of  the  preceding  circumstances 
leads  to  the  conclusion,  that  the  moon  is  an  opaque 
globular  body,  having  its  surface  diversified  with 
mountains  and  vallies;  and  that  it  shines  by  reflecting 
the  light  of  some  other  body.  And  from  the  relative 
situations  of  the  sun,  moon,  and  earth,  at  the  times  of 
the  different  phases,  this  body  appears  to  be  the  sun. 

At  the  time  of  new  moon,  the  sun  and  moon  are  in 
nearly  the  same  direction  from  the  earth,  and  as  the 
moon’s  distance  is  less  than  the  sun’s  (5.14  and  16), 
the  enlightened  side  of  the  moon  is  turned  from  the 
earth,  and  it  is  therefore  invisible.  At  the  first  or  last 
quarter,  the  difference  of  longitudes  of  the  sun  and 
moon  being  90°  or  270°,  it  is  plain  that  about  one  half 
of  the  enlightened  side  is  turned  towards  the  earth.  At 
the  full,  the  moon  being  in  opposition  to  the  sun,  near¬ 
ly  its  wrhole  enlightened  side  is  turned  towards  the 
earth. 


96 


ASTRONOMY. 


47-  If*  at  the  time  of  new  moon,  the  moon  is  at  or 
near  to  one  of  its  nodes,  as  it  is  an  opaque  body,  and 
is  then  directly  between  the  sun  and  earth,  or  very 
nearly  so,  it  must'  prevent,  at  least,  some  of  the  sun’s 
light  from  arriving  at  the  earth.  This  is  sometimes  the 
case,  and  occasions  what  is  called  an  Eclipse  of  the 
Sun.  When  $4  the  time  of  full  moon,  the  moon  is  at, 
or  near  the  node,  it  is  in  the  earth’s  shadow,  and  an 
Eclipse  of  the  Moon  is  produced.  When  the  moon 
passes  between  the  earth  and  a  star  or  planet,  the  cir¬ 
cumstance  is  called  an  Occultation  of  the  star  or  pla¬ 
net.  These  phenomena  will  be  considered  in  the  next 
chapter. 

18.  The  Elongation  of  a  body  is  its  angular  dis¬ 
tance  from  the  sun,  as  seen  from  the  earth. 

19.  The  apparent  breadth  of  the  visible,  enlightened 
part  of  the  moon,  is  nearly  equal  to  the  apparent  di¬ 
ameter,  multiplied  by  the  square  of  the  sine  of  the 
elongation. 


Let  E,  Fig.  23,  be  the  centre  of  the  earth,  MANB  the  moon, 
C  its  centre,  and  let  the  sun  be  in  the  direction  CS.  Also  let 
ACB  be  perpendicular  to  CS,  AD  to  EC,  and  EM,  EN,  tan¬ 
gents  to  the  moon  at  M  and  N.  As  this  is  not  a  problem  in  which 
it  is  important  to  obtain  great  accuracy,  we  may  suppose  the 
moon  to  be  in  the  plane  of  the  ecliptic,  and  the  rays  of  light  from 
the  sun,  that  arrive  at  the  moon,  to  be  parallel.  Then  AMB  is  the 
enlightened  half  of  the  moon,  of  which  only  the  part  AM  is  visi¬ 
ble  from  the  earth.  It  is  seen  under  the  angle  AExM. 

Put  C  =  angle  ECS, 

£  =  angle  CEM  =  app.  semidiam.  of  moon. 

Then  CD  =  AC  cos  ACE  =  AC  sin  SCE  =  AC  sin  C, 
ED  =  EC  —  AC  sin  C, 

AD  =  AC  sin  ACE  =  —  AC  cos  SCE  =  —  AC  cos  C. 


AC  cos  C 
EC  —  AC  sin  C 


tan  AEC 


AD 

ED 


CHAPTER  X. 


97 


But  AC  =  EC  sin  *(7.13). 

EC  sin  *  cos  C 


Hence  tan  AEC  =  — 
sin  *cos  C 


EE— EC  sin  *sinC 


1  —  sin  *  sin  C 

Or  since  AEC  and  *  are  both  small, 

*  cos  C 


AEC  = 


1  —  sin  *  sin  C 
AEM  =  CEM  —  AEC  =  * 


cos  C 


_  }  4.  S'  cos  C  —  *  cos  C  -f 
_  }  -f  *  cos  C  4. 


1  — sin  *  sin  C 
*  cos  C 


1  —  sin  *  sin  C 
sin  *cos  C  sin  C 


1  —  sin  P  sin  C 


As  the  last  term  is  very  small  it  maybe  rejected,  and  we  shall 
have, 

AEM  =  *  -f  *  cos  C  =  (1  -f  cos  C)  =  2  *  cos2  C  (App. 

9) . A. 

On  account  of  the  small  distance  of  the  moon  from  the  earth, 
compared  with  the  sun’s  distance  from  both  of  them  (5.14  and 
16),  we  may,  without  material  error  in  the  present  case,  consider 
the  line  ES',  drawn  from  the  earth  in  the  direction  of  the  sun,  as 
parallel  to  CS.  Then, 

AEM  =  *  4.  *cos  C  =  *  +  *cos  (180°  —  CES') 

=  * —  *  cos  CES' s  *.  (1  —  cos  CES') 

==  2  *  sin2  \  CES',  (App.  8)  B. 

20.  In  like  manner  as  the  moon  reflects  light  to  the 
earth,  the  earth  reflects  light  to  the  moon,  with  this 
difference,  that  when  the  moon  gives  least  light  to  the 
earth,  the  earth  gives  the  most  to  the  moon,  and  the 
contrary.  It  is  the  light  reflected  from  the  earth  to 
the  moon,  and  from  it  back  to  the  earth,  that  renders 
the  obscure  part  visible. 

14 


I 


98  ASTRONOMY. 

moon’s  MEAN  MOTION  IN  LONGITUDE. 

21.  If  the  time  at  which  the  moon’s  centre  passes 
the  meridian  be  observed,  its  right  ascension  at  that 
times  becomes  known.  And  from  the  observed  alti¬ 
tude,  corrected  for  refraction  and  parallax,  the  north 
polar  distance  and  consequently  the  declination  is  ea¬ 
sily  obtained  (6.2).  With  the  right  ascension  and  de¬ 
clination,  the  longitude  may  be  calculated  (6.19).  By 
repeated  observations  and  calculations  of  this  kind  the 
interval  from  the  time  at  which  the  moon  has  any  given 
longitude,  till  it  again  arrives  at  the  same  longitude, 
may  be  ascertained.  This  interval,  which  is  the  Tro¬ 
pical  Revolution  of  the  moon,  is  found  to  vary.  From 
observations,  made  at  distant  periods,  its  mean  length 
is  determined  to  be  27.321582  mean  solar  days.* 
Hence  27-321582  d.  :  1  d.  :  :  360°  :  13°  10'  35"  = 
moon’s  daily  mean  motion  in  longitude. 

The  mean  motion  of  the  moon,  here  given,  corres¬ 
ponds  to  the  commencement  of  the  present  century.  A 
comparison  of  modern  observation,  with  those  of  re¬ 
mote  periods,  proves  that  the  moon’s  mean  motion  is 
subject  to  a  small,  but  thus  far  a  continual  Acceleration . 
Investigations  in  Physical  Astronomy,  have  made 
known  the  cause  of  this  acceleration,  and  have  shown 
that  it  is  really  a  periodical  inequality  in  the  moon’s 
motion,  which  requires  a  great  length  of  time  to  go 
through  its  different  values. 

22.  The  moon’s  mean  longitude,  the  1st  of  January 
1801  at  mean  noon,  on  the  meridian  of  Greenwich,  was 
3s  28°  16'  56".  1.  With  the  j Epoch  of  the  mean  longi¬ 
tude,  and  the  moon’s  mean  motion,  the  mean  longitude 
may  be  determined  for  any  other  given  time. 

*  It  may  be  observed  that  astronomers  generally  express  such  periods  in 
mean  solar  time;  unless  the  contrary  is  specified. 


CHAPTER  X. 


99 


moon’s  nodes  and  inclination  of  the  orbit. 

23.  With  different  longitudes  and  latitudes  of  the 
moon,  deduced  from  the  observed  right  ascensions  and 
declinations,  the  situations  of  the  nodes  and  the  incli¬ 
nation  of  the  orbit  may  be  found,  by  methods  nearly 
similar  to  those  employed  for  determining  the  place  of 
the  vernal  equinox,  and  the  obliquity  of  the  ecliptic 
(7.6  and  6.2). 

24.  The  moon’s  nodes  are  not  fixed;  they  have  a 
retrograde  motion,  which  is  ascertained,  by  determin¬ 
ing  their  situations  at  different  periods.  This  mo¬ 
tion  is  subject  to  several  inequalities,  and  diminishes 
from  century  to  century.  At  the  commencement  of  the 
present  century  the  mean  length  of  a  tropical  revolution 
of  the  nodes,  was  6798.54019  days  or  18  years  and 
224.17944  days;  and  at  that  time  the  mean  longitude 
of  the  ascending  node  was  0s  13°  52'  47".  The  annual 
mean  motion  of  the  nodes  in  longitude  is  19°  SO'  26". 

25.  The  inclination  of  the  moon’s  orbit  to  the  plane 
of  the  ecliptic,  varies  from  about  5°  to  5°  18'.  It  is 
greatest  when  the  moon  is  in  quadratures  and  least 
when  it  is  in  syzigies.  The  mean  inclination,  which 
is  always  the  same,  is  5°  8'  38". 

moon’s  orbit. 

26.  Observations  and  investigations,  similar  to  those 
made  in  the  case  of  the  sun,  prove  that  the  moon’s  orbit 
is  nearly  an  ellipse,  having  the  centre  of  the  earth  in 
one  focus,  about  which  the  radius  vector  describes 
areas  nearly  proportional  to  the  times. 

27.  Considering  the  mean  distance  of  the  moon  from 
the  earth,  a  unit,  or  1,  the  eccentricity  of  the  orbit  is 
.0548553,  which  gives  the  greatest  equation  of  the  cen¬ 
tre  6°  17'  54". 5.  " 


100 


ASTRONOMY. 


28.  The  apsides  of  the  moon’s  orbit;  have  a  direct 
motion;  by  which  they  perform  a  mean  tropical  revo¬ 
lution  in  3231.' 4731  days,  or  8  years  and  309.537 
days.  At  the  commencement  of  the  present  century, 
the  mean  longitude  of  the  perigee  was  8s  26°  10'. 

29.  Besides  the  ecpiation  of  the  centre;  the  moon’s 
motion  is  subject  to  numerous  inequalities,  of  which 
the  three  following  are  the  principal. 

The  most  considerable  is  called  the  Erection,  and 
was  discovered  by  Ptolemy.  It  depends  on  the  angu¬ 
lar  distance  of  the  moon  from  the  sun,  and  of  the  moon 
from  the  perigee,  and  amounts,  when  greatest,  to  1° 
20'  30". 

The  second  is  called  the  Variation ,  and  was  disco¬ 
vered  by  Tycho  Brahe.  It  disappears  when  the  moon 
is  in  the  syzigies  and  quadratures,  and  is  greatest 
when  it  is  in  the  octants.  It  then  amounts  to  35'  42". 

30.  The  third  is  called  the  Annual  Equation ,  and 
depends  on  the  mean  anomaly  of  the  sun.  When 
greatest,  it  amounts  to  11'  12". 

30.  The  motion  of  the  moon’s  nodes,  the  change  in 
the  inclination  of  the  orbit,  the  motion  of  the  apsides, 
and  the  preceding  inequalities  in  the  moon’s  motion, 
are  caused  by  the  sun’s  attraction,  and  are  completely 
explained  by  investigations  in  physical  astronomy. 
These  investigations  have  also  led  to  the  discovery  of 
other  minute  inequalities  in  the  moon’s  motion,  and 
thence  conduced  to  the  accuracy  of  tables  for  computing 
its  place  at  any  given  time. 

31.  The  most  accurate  tables  of  the  moon  are  those 
by  Burg,  and  by  Burckhardt.  The  former  employs 
28  equations  for  the  moon’s  longitude,  and  the  latter  36. 
The  moon’s  place  calculated  by  either  of  these  sets  of 


CHAPTER  X. 


101 


tables,  always  agrees,  within  a  few  seconds,  with  its 
place  as  determined  by  observation. 

DIFFERENT  REVOLUTIONS  OF  THE  MOON. 

32.  From  the  tropical  revolution  of  the  moon  (21), 
by  taking  into  view  the  motion  of  the  equinoxes,  the 
sun’s  apparent  motion,  and  the  motions  of  the  apsides, 
and  nodes  of  the  moon’s  orbit,  the  other  revolutions  of 
the  moon  are  easily  determined.  The  following  are 
the  lengths  of  these  revolutions,  as  given  by  Delambre: 

DAYS. 

Tropical  revolution,  -  -  -  27-3215255 

Sidereal  revolution,  -  27-3215830 

Synodic  revolution,  -  -  -  29-5305885 

Anomalistic  revolution,  -  -  27-5515704 

He  volution  from  one  node  to  the  same,  27.2122222 

moon’s  revolution  on  its  axis. 

33.  It  has  been  stated  (15),  that  the  moon  presents 
nearly  the  same  face  to  the  earth.  It  must  therefore 
revolve  on  its  axis  in  the  same  direction  and  same  time 
that  it  revolves  in  its  orbit.  But  accurate  and  continued 
observations,  show  that  the  moon’s  spots  do  not  pre¬ 
serve  exactly  the  same  situations  on  the  disc.  They 
are  seen  alternately  to  approach  and  recede  from  the 
edge.  Those  that  are  very  near  the  edge,  successively 
disappear  and  again  become  visible,  making  periodical 
oscillations,  which  are  called  the  Librations  of  the 
moon. 

34.  The  librations  of  the  moon  are  not  occasioned 
by  an  unequal  motion  on  its  axis.  For,  admitting  this 
motion  to  be  uniform,  and  the  axis  to  be  nearly  per¬ 
pendicular  to  the  plane  of  the  orbit,  small  portions  on 


102 


astronomy. 


the  east  and  west  sides  of  the  moon  ought  alternately 
to  come  into  sight  and  to  disappear,  in  consequence  of 
its  irregular  motion  in  its  orbit.  This  is  conformable 
to  observation,  and  is  called  the  Libration  in  Longi¬ 
tude. 

35.  Besides  the  motion  of  the  spots  in  an  easterly 
and  westerly  direction,  they  are  observed  to  have  a 
small  alternate  motion  from  north  to  south.  This  is 
called  the  Libration  in  Latitude ,  and  shows  that  the 
moon’s  axis,  though  nearly,  is  not  exactly,  perpendicu¬ 
lar  to  the  plane  of  its  orbit. 

36/  In  consequence  of  the  earth’s  diurnal  motion,  a 
spectator  at  the  surface  sees  portions  of  the  moon  a 
little  different,  according  to  its  different  positions  above 
the  horizon.  This  is  called  the  Diurnal  or  Parallactic 
Libration. 

37.  From  calculations  founded  on  accurate  observa¬ 
tions  of  the  lunar  spots,  it  has  been  found,  that  the 
equator  of  the  moon  is  inclined  to  the  plane  of  the 
ecliptic  in  an  angle  of  1°  30';  and  that  the  line,  in 
which  the  plane  of  the  equator  cuts  the  plane  of  the 
ecliptic,  is  always  parallel  to  the  line  of  the  nodes. 

38.  If  three  planes  be  supposed  to  pass  through  the 
centre  of  the  moon,  one  representing  the  equator  of  the 
moon,  another  the  plane  of  its  orbit,  and  the  third  be¬ 
ing  parallel  to  the  ecliptic;  then  the  last  will  lie  between 
the  two  others,  and  will  intersect  them  in  the  same  line, 
in  which  they  intersect  each  other.  It  will  make  with 
the  first,  an  angle  of  1°  30';  and  with  the  second,  an 
angle  of  5°  9'.  This  curious  fact  was  discovered  by 
Cassini;  and  it  has  been  explained  by  Lagrange  from 
the  theory  of  gravity. 


CHAPTER  X. 


103 


moon’s  diameter  and  mountains. 


39.  The  greatest  and  least  horizontal  parallaxes  of 
the  moon,  for  a  place  on  the  equator  are  53'  52"  and 
6l'  32"  (5.14).  The  corresponding  apparent  diame¬ 
ters  are  found  to  be&9'  22".  and  33'  31".  If  d  =  real 
diameter  of  the  moon  and  R  =  equatorial  radius  of 
the  earth,  we  have  (7-15), 


7  _  t>  29'  22"  _  R  1762 

53'  52"  *  3230 


2U. 


1762 

6460 


2R.  —  very  nearly. 
11 


Hence  the  diameter  of  the  moon  is  about  T3T  of  the 
equatorial  diameter  of  the  earth,  and  consequently  its 
surface  is  about  T?  of  the  earth’s  surface,  and  its  vol¬ 
ume,  about  -1-  of  the  earth’s  volume.  The  moon’s  di¬ 
ameter  in  English  miles  is  2160. 

40.  It  has  been  observed  (16)  that  the  moon’s  sur¬ 
face  is  diversified  with  mountains  and  vallies.  The 
heights  of  some  of  these  mountains,  in  comparison  with 
the  diameter  of  the  moon,  are  found  to  exceed  those  of 
the  earth.  Though  not  a  subject  of  much  importance, 
it  may  be  interesting  to  the  student  to  know  a  method 
of  ascertaining  the  heights  of  the  lunar  mountains. 

41.  Let  ABO,  Fig.  24,  be  the  enlightened  hemisphere  of  the 
moon,  E  the  situation  of  the  earth,  ES'  the  direction  of  the  sun 
from  the  earth,  and  SM  a  solar  ray,  touching  the  moon  in  0, 
which  will  be  one  of  the  points  in  the  curve,  separating  the  en¬ 
lightened,  from  the  dark  part  of  the  moon.  Also  let  M  be  the 
summit  of  a  mountain,  situated  near  to  0,  and  sufficiently  eleva¬ 
ted  to  receive  the  sun’s  light.  To  a  spectator  at  E,  the  summit 
M  of  the  mountain,  will  appear  as  a  bright  spot  on  the  dark  part 
of  the  moon. 

The  angle  MEO  may  be  measured  by  means  of  a  micrometer, 


104 


ASTRONOMY, 


attached  to  the  telescope.  In  this  case  as  in  a  former  (19),  we 
may  consider  ES'  as  parallel  to  MS.  We  may  also  without  ma¬ 
terial  error  consider  the  angle  MES'  as  equal  to  the  elongation 
CES',  Let  OD  be  perpendicular  on  ME,  and  put, 

^  =  apparent  diameter  of  the  moon, 
d  =  AO  z—  real  diameter  of  the  moon. 

OD  EO  sin  MEO 


We  have,  CO  tan  MCO  =  MO  = 


sin  OME 


EO  sin  MEO 
sin  MES' 

Hence,  tan  MCO 


EC  cin  MEO 


sin 

CES' 

EC 

sin  MEO 

CO* 

sin  CES' 

2 

sin  MEO 

nearly. 


sin 


sin  OME 


sin  MEO 
sin  CES' 
(7.13). 


sin^  sin  CES' 


Height  of  the  mountain  —  a  M  =  MC  —  OC 


OC 


=  OC. 
=  OC. 


1 


cos  MCO 


sin  MCO 
1  — -  cos  MCO 


OC. 


cos  MCO 
1  —  cos  MCO  sin  MCO 


OC 


cos  MCO 


sin  MEO 


tan  MCO  =  OC. 


1 


cot  h  MCO 


sin  MCO 
tan  MCO 


(App.  1 1). 


—  OC  tan  MCO  tan  £  MCO  =  h  OC  tan2  MCO 
_  A  0G  4  sin2  MEO  =  d  /  sin  MEO  \ 2 
— "  2  *  sin2  /  sin2  CES'  *  \sin  ^  sin  CES'/ 

.  /  ang.  MEO  \  2 

=  *  U  sin  CES'/  ’ 


Dr.  Herschel  has  made  observations  on  a  number  of  the  lunar 
mountains.  For  one  of  these  the  data  are,  the  angle  MEO  = 
40". 625,  apparent  diameter  of  the  moon  =  32'  5"  2,  and  the 
elongation  =  125°  8'.  Hence  taking  the  moon’s  diameter  2160 
miles,  we  easily  obtain  from  the  preceding  formula,  the  height  of 
the  mountain  =  1.45  miles. 

42.  Luminous  spots,  which  are  entirely  unconnected 
with  the  phases,  or  in  other  words,  are  not  the  reflec- 


CHAPTER  X.  403 

lion  of  the  sun’s  light,  are  sometimes  seen  on  the  moon’s 
disc.  These  are  supposed  to  be  volcanoes. 

43.  If  the  moon  was  surrounded  with  an  atmosphere, 
such  as  appertains  to  the  earth,  it  would,  by  its  action 
in  changing  the  rays  of  light,  produce  a  very  sensible 
effect  in  the  duration  of  an  eclipse  of  the  sun,  or  an  oc- 
cultation  of  a  star  or  planet  (17).  But  various  accu¬ 
rate  observations  prove  that  if  any  effect  of  this  kind 
has  place,  it  is  extremely  small.  It  therefore  follows, 
that  if  the  moon  has  any  atmosphere,  it  must  be,  either 
very  limited,  or  very  rare. 

moon’s  passage  over  the  meridian. 

44.  In  consequence  of  the  moon’s  daily  increase  in 
right  ascension,  it  passes  the  meridian  later  on  each 
day  than  on  the  preceding  (2).  The  daily  retardation 
varies  from  about  38,  to  66  minutes. 

To  obtain  the  time  of  the  moon’s  passage  on  any  particular 
day,  let  R  =  the  excess  of  the  moon’s  right  ascension  above  the 
sun’s,  at  noon  of  the  given  day,*  S  =  daily  motion  of  the  sun  in 
right  ascension,  M  =  that  of  the  moon,  both  being  considered  as 
uniform  during  the  day,  T  =  the  required  time  of  the  moon’s 
passage,  and  A  =  the  arc  of  the  equator,  which  passes  the  meri¬ 
dian,  between  noon  and  the  moon’s  passage. 

T.M 

Then24h  :  T  : :  M  :  =  moon’s  motion  in  right  ascension 

during  the  time  T. 

Hence  A  =  R  -f  — 

24“ 

Also  24"  :  T  : :  360’  +  S  :  A  =  T-  (360°  +  S) 

24" 

*  When  the  moon’s  ascension  is  less  than  the  sun’s,  it  must  be  increased 
by  360°  or  24  h. 


10G 


ASTRONOMY. 


Therefore  =  R  +  ™ 

24‘*  24h 

T.  (36(b  4-  S)  =  24h.  R  +  T.  M, 
24h.  R  24h.  R 

“  360°  +  S  —  M  “  24h  +  S  —  M* 


45.  If  M  =  the  daily  motion  of  a  planet  in  right  ascension,  the 
preceding  formula  will  give  the  time  of  its  passage  over  the  me¬ 
ridian,  observing  that  when  the  motion  of  the  planet  is  retrograde, 
the  sign  of  M  must  be  changed  and  the  formula  will  then  become, 

T  =  24h  R 

360°  +  S  +  M’ 

For  a  fixed  star,  M  =  0,  and  the  formula  becomes, 

24u  R 


360° 


+ 


moon’s  rising  and  setting. 

46  On  account  of  the  moon’s  change  in  declination,  the  semi¬ 
diurnal  arc,  found  with  its  declination  at  the  time  of  its  passing 
the  meridian  (9.18),  is  not  correct.  If  however  the  semi¬ 
diurnal  arc,  thus  obtained,  be  applied  to  the  time  of  the  moon’s 
passage,  by  subtracting  for  the  rising  or  adding  for  the  setting,  it 
will  give  the  approximate  time  of  rising  or  setting. 

To  obtain  the  time  more  correctly,  find  the  declination  for  the 
approximate  time  and  again  calculate  the  semi-diurnal  arc,  which 
must  be  corrected  on  account  of  the  moon’s  change  in  right  as¬ 
cension.  Thus,  24h  -f  S  —  M  :  24h  : :  semr-diurnal  arc  :  cor¬ 
rected  semi-diurnal  arc,  which  applied  as  before  to  the  time  of 
the  moon’s  passage,  gives  the  time  of  the  moon’s  rising  or  setting, 
very  nearly. 

If  D  =  the  difference  between  the  times  of  the  moon’s  passage 
on  two  consecutive  days,  one  of  which  precedes  and  the  other  fol¬ 
lows  the  required  time  of  the  moon’s  rising  or  setting,  the  last 
correction  may  be  made  thus  :  As  24h  :  24h  +  D  :  :  semi¬ 
diurnal  arc  :  corrected  semi-diurnal  arc. 

As  the  mean  length  of  the  semi-diurnal  arc  is  about  6  hours; 
it  is  better,  in  the  operation  for  obtaining  the  approximate  time  of 


CHAPTER  X. 


107 

rising  or  setting,  to  make  use  of  the  moon’s  declination  6  hours 
before  or  after  its  passage  over  the  meridian,  according  as  it  is 
the  rising  or  setting  that  is  required. 

moon’s  parallax  in  longitude  and  latitude. 

47*  The  effect  of  parallax  in  changing  the  altitudes 
of  the  heavenly  bodies  has  been  shown  in  a  preceding 
chapter  and  a  method  given  for  determining  it.  But 
this  is  not  the  only  effect  of  parallax.  It  also  changes 
the  right  ascension,  declination,  longitude,  and  lati¬ 
tude  of  a  body.  It  may  be  proper  here  to  investigate 
formulae  for  calculating  the  effect  of  parallax  on  the 
moon’s  longitude  and  latitude,  as  they  will  be  useful 
in  our  chapter  on  eclipses  of  the  sun. 

48.  Let  HZR,  Fig.  25,  be  the  meridian,  HR  the 
horizon,  Z  the  zenith,  EQ  the  equator,  ECO  the  eclip¬ 
tic,  P  and  p  their  poles,  E  the  vernal  equinox,  A  the 
true  place  of  the  moon,  B  its  apparent  place,  as  de¬ 
pressed  by  parallax,  in  the  verticle  circle  ZABK,  and 
pAa  and  pftb,  circles  of  latitude,  passing  through 
the  true  and  apparent  places.  By  the  effect  of 
parallax,  the  true  longitude  Ea  is  changed  to  the  ap¬ 
parent  longitude  E  b,  and  the  true  latitude  A  a,  to  the 
apparent  latitude  B b. 

49.  The  difference  between  the  true  and  apparent 
longitude  of  a  body,  produced  by  parallax,  is  called 
the  Parallax  in  Longitude;  and  the  difference  between 
the  true  and  apparent  latitude,  is  called  the  Parallax 
in  Latitude. 

50.  The  point  in  the  ecliptic,  which  is  above  the  ho¬ 
rizon,  and  at  90°  distance  from  the  intersection  of  the 
ecliptic  and  horizon,  is  called  the  JSTonagesimal  De¬ 
gree  of  the  Ecliptic . 

51.  The  data  usually  given  to  calculate  the  moon’s 


108 


ASTRONOMY. 


parallax  in  longitude  and  latitude,  are  the  moon’s  true 
longitude,  and  its  latitude,  or  distance  from  the  north 
pole  of  the  ecliptic,  its  horizontal  parallax,  the  obliqui¬ 
ty  of  the  ecliptic,  the  latitude  of  the  place,  and  the  right 
ascension  of  the  mid -heaven. 

On  account  of  the  spheroidical  figure  of  the  earth, 
the  horizontal  parallax,  at  any  given  time,  is  different 
at  different  places  (5. 12).  The  parallax  for  a  given 
place,  is  called  the  Reduced  Parallax .  If  the  earth 
were  a  sphere,  having  a  radius  equal  to  the  straight 
line  joining  the  given  place  and  the  centre,  it  is  evident 
that  the  parallax  for  all  parts  of  it,  would  be  the  same 
as  the  reduced  parallax  for  the  given  place.  It  is  also 
plain,  from  the  definition  (4.10),  that  the  reduced  lati¬ 
tude  of  any  place,  is  its  latitude  on  the  supposition  of 
the  earth  being  a  sphere.  If,  therefore,  the  reduced 
latitude  and  parallax  be  used,  the  earth  may  be  con¬ 
sidered  as  a  sphere,  which  will  simplify  the  investiga¬ 
tions  for  finding  the  parallax  in  longitude  and  latitude. 

52.  Let  pPCD  be  a  great  circle,  passing  through  p  and  P,  and 
pZn I  another,  passing  through  p  and  Z.  Because  P  is  the  pole 
of  EQ,  the  pole  of  pPD  is  in  EQ;  and  because  p  is  the  pole  of 
EC,  the  pole  ofpPD  is  in  EC;  the  point  E,  in  which  EQ  and 
EC  intersect  each  other,  is,  therefore,  the  pole  ofpPD;  and  con¬ 
sequently,  ED  and  EC  are  each  90°.  In  like  manner,  because 
Z  and  p  are  the  poles  of  HR  and  EO,  the  point  0,  is  the  pole  of 
pZnl,  and  On  and  01  are  each  90°.  Consequently,  n  is  the  nona- 
gesimal  degree  of  the  ecliptic.  Also,  E n  is  the  longitude  of  the 
nonagesimal  degree,  and  nl  is  its  altitude.  These  quantities  are 
used  in  finding  the  parallax  in  longitude  and  latitude,  and  must 
first  be  found. 

53.  Put 

=  P p  —  obliquity  of  the  ecliptic, 

H  =  ZP  =  complement  of  the  reduced  latitude, 


CHAPTER  X. 


109 


M=  EM  ==  right  ascension  of  the  mid-heaven, 

n  .==  E n  =  longitude  of  the  nonagesimal, 

h  —  pZ  —  90°  —  Zn  —  nl  =  altitude  of  the  nonagesimal, 

L  =  Ea  =  moon’s  true  longititude, 

A  =  pA  =  moon’s  true  distance  from  north  pole  of  the  ecliptic, 
33-  —  moon’s  horizontal  parallax,  reduced, 
n  =  moon’s  parallax  in  longitude, 

7T  =  moon’s  parallax  in  latitude. 

In  the  triangle  pPZ,  we  have  given  Pp,  PZ,  and  pPZ  = 
180°  ZPD  =  180°  —  DM  =  180°  —  (90°  —  EM)  =  90° 
+  EM  =  90°  -f  M,  to  find  pZ  —  the  altitude  of  the  nonagesi¬ 
mal,  and  PpZ  =  Cn  =  90°  —  E n  =  the  complement  of  the 
longitude  of  the  nonagesimal. 

LONGITUDE  OF  THE  NONAGESIMAL. 


Let  S  =  PpZ  +  PZp,  D  =  PpZ  —  PZp,  E  =  180°  —  *  S, 
and  F  =  180°  —  |  D.  Then  (App.  41), 


tan  |  S  == 


cos  -i  (H  —  *) 


cos  -J 


cos 


(H 

(H 


+  *) 
—  a) 


tan (180° 


|S)= 


cos 

cos 


(H 


+  ") 
—  *) 


cos 

cos 


(H 

(H 


+ 


cos  4  (H  +  ») 
__  cos  (H  —  *>) 
COS  (H  4-  a) 

or  tan  E  =  cos  1 (H  Hi! 

COS  4  (H  +  a) 


.  cot  ‘  (90°  +  M) 

.  tan  4  (90°  —  M), 

.  tan  (180°  —  45°  +  \  M) 
,  tan  4  (270°  +  M) 

.  tan  4  (M  —  90°), 

.  tan  4  (M  —  90°). 


Again  (App.  42), 


tan  4  D  =  cot  '  (90°  +  M), 

sin  4  (H  +  *) 


Whence,  by  transforming  as  above,  we  have, 
tan 


F  =  sin  %  (H~  tan  t  (M  —  90°). 
sin  |  (H  4-  *) 


110 


ASTRONOMY. 


Now  IS  +  |D  =  VpZ  =  90°  —  n. 

Hence,  n  =  90°  —  (4  S  -f  4  D), 
or  w  =  450°  —  (i  S  4-  4  D)  =  180°  —  4  S  +  180°  — 
i  D  +  90°. 

Consequently,  n  =  E  -f  F  -f  90°,  rejecting  360',  when  the 
sum  exceeds  that  number. 

ALTITUDE  OF  THE  NONAGESIMAL. 


We  have  (App.  44), 


tan  ih  =  s-jn.  1  S.  tan  i  (H  —  ») 


sin  3  D 
sin  3  E 
sin  4  F* 


tan  4  (H  —  *). 


(5.5). 


PARALLAX  IN  LONGITUDE. 

54.  The  triangle  ApB,  gives, 

.  „  •  »  t>  sin  AB  sin  ZBp 

sin  n  ==  sin  ApB  = - L 

sin  A p 

_  sin  w  sia  (N  4.  p )  sin  ZB p 
sin  A 

In  the  triangle  ZpB, 

the  angle ZpB=nb=na  4-  ab=  Ea — En  +  cib—h — n  -f  n, 

and  sin  ZB»  — sin  PZ  sin  ZPB  _  sin /i  sin  (L — n  +  n) 
sin  ZB  sin  (N  4-  p) 

tt  ♦  rr  sin®- sin  h  .  ,T  ,  _v  /ri. 

Hence  sin  n  —  — . — . —  sm  (L  —  n  4-  n)  (C). 


sin  A 


sin  A 


sin  n  =  sin  (L  —  n  +  n)  =  sin  (L  —  n)  cos  n 
—  n)  sin  n 

=  sin  (L  —  n)  cot  n  4.  cos  (L  —  n) 


sin  w  sin  h 
4-  cos  (L  —  n)  sin  n 
sin  A 

sin  w  sin  h 
cot  n 


sin  A 


cos  (L  —  n) 

sin  w  sin  h  sin  (L  —  n)  sin  (L  —  n ) 

Make  tan  «  =  sin  >  sin  h  sin  (L  -  »)  =  fesin(L-n). 

sin  A  sm  A 


CHAPTER  X. 


lit 


Then,  - 


sm  A 


sin  »■  sin  h  sin  (L  —  ri) 


.  cos  it 

—  cot  it  =  - - , 

sin  it 


and  cot  n  = 


cos  w  cos(L — ri) 


sin  it  sin  (L  —  ri) 
sin  (L  —  ri)  cos  it —  cos(L  —  ri)  sin  it  __  sin(L  —  n  —  u ) 
sin  (L  —  n)  sin  it  sin  ( L  —  ri)  sin  u 

Hence,  tan  n  =  — sin  w. 

sm(L — n — u ) 


PARALLAX  IN  LATITUDE. 

65.  The  triangles  pZA  and  pZB  give  (App.  34), 

cos  ZB  —  cos  ^  —  cos  cos  ^  —  cos  — cos  cos  ^ 

sin  Zp  sin  ZA  sin  Zp  sin  ZB 


or, 


cos  A  — cos  h  cos  N  cos  (A  4.  *■)  —  cos  h  cos  (N  4-  p ) 


sin  N  sin  (N  4-  p) 

cos  A  sin  (N  +  p)  —  cos  h  cos  N  sin  (N  -f  p)  =  cos  (A  -f  ri)  sin 
N  —  cos  h  sin  N  cos  (N  4  p) 

cos  (A  4.  «r)  sin  N  =  cos  A  sin  (N  4 -  p)  —  cos  h  [sin  (N  4-  p) 
cos  N  —  cos  (N  4-  p)  sin  N] 

=  cos  A  sin  4-  p)  —  cos  h  sin  p  (App.  13) 

—  cos  A  sin  (N  4.  p')  —  cos  h  sin  zr  sin  (N  4-  p)  (5.5) 
=  sin  (N  4-  p).  (cos  A  —  sin  w  cos  h.) 

Therefore,  cos  (A  4-  n)  =  j Z.P),  (cos  A  —  sin  cos  h). 

sin  N 

But,  sin  AP  sin  ZPA  =  s;n  pzb  =  sinBP8inZPB 
sin  ZA  ^  sin  ZB 

Qr  sin  A  sin  (L  —  n)  __  sin  A  4  sin  (L  —  n  - f-  n) 
sin  N  sin  (N+p; 

sin  (N  4  p)  __  sin  (A  4  gr>  sin  (L  — n  4-  n) 
sin  N  sin  A  sin  (L  —  n) 

Hence, cos(A  +  *)  =  >in.(A.+  y)"°  (L~n  +  ?> 


sin  w  cos  h) 


sin  A  sin  (L  —  ri) 


m 

.  (cos  A  — 


j  s  a  1  x  sin  (L — n  4-  n)  ,  *  sin^-cos/tx  /t,x 

cot  (A  +  ri)  =  —4 - - - (cot  A - - - ),  (E) 


sin  (L  —  ri) 


sin  A 


112 

Make  tan  x  = 
Then,  cot  (A  4-  *) 


ASTRONOMY. 

sin  w  co?  /t 
sin  A 

sin  (L —  n  -f-  n) 
sin  (L  — w) 

sin  (L — n  4-  n)  ^cosA 


(cot  A  —  tan  x ) 


sin  x 
cos 


;) 


sin  (L  —  n)  '  sin  A 

_ sin  (L  —  n  4  n)  cos  A  cos  x  —  sin  A  sin  z 

sin  (L  —  n)  sin  A  cos  x 

__  sin  (L  —  n  4-  n)  cos  (A  4-  x ) 


sin  (L  —  n)  sin  A  cos  x 


(.n 


The  apparent  latitude  (A  4-  *■)  being  calculated  by  either  of 
the  formulae  (E)  and  (F),  we  have  ^  =  (A  4-  w)  —  A. 

56  We  may  obtain  formulae  that  will  give  the  parallax  in  la¬ 
titude,  without  first  finding  the  apparent  latitude,  and  which  in 
some  cases  may  be  more  convenient  than  those  in  the  last  article. 

We  have  (55.E) 

sin  (L  —  n)  cot  (A  4-  tt) _  ^ _  sin  a*  cos  h 

sin  (L  —  n  - j-  n)  sin  A 

cot  a  —  s*n  eos  ^  cot  (A  4-  ,r) 

sin  A  sin  (L  —  n  4-  rr) 

1  f  a  ,  x  sin  a- cos  1l  .  x 

cot  A  —  cot  (A  4-  yr)  —  - —  cot  ( A  4.  *)  _l 

'  sin  A  ^ 

sin  (L  —  ri)  cot  (A  4.  «■) 

sin  (L  —  n  4-  n) 

Hence  (App.  27), 


sin  tt 


sin  a-  cos  h 


sin  A  sin  (  A  4-  x)  sin  A 

cot  (A  4-  t)  [sin  (L —  n  4.  n)  —  sin(L —  n)J 
sin  (L  —  w  4-  n) 

__  sin  -zr  cos  h  _  2  sin  §  n  cos  (L  —  n  4-  h  n)  cot  (A  4-  *•) 

sin  A  sin  —  n  4-  n j 

(App.  21) 

sin  5r  —  sin  a*  cos  h  sin  (A  4  »)  — 


CHAPTER  X. 


113 


2  sin  2  II  sin  A  cos  (L  —  »i  +  |n)  cos  (A  4.  w) 


sin  (L  —  n  4-  n) 

=s  sin  w  cos  h  sin  (A  4.  «■) 

sin  n  sin  A  cos  (L  —  n-|-|n)  cos  (A  4-  «•) 
cos  2  n  sin  (L  —  n  +  n) 


(APP.  7) 


But  (54. C),  — sln_ A —  —  sin  ®-  sin  h.  Therefore, 
^  sin  (L  —  n  4-  n)  5 

sin  «r  =  sin  ar  cos  h  sin  (A  f  w) 

sin  73-  sin  h  cos  (L  — n  +  in)  cos  (A  4-  *■) 

cos  2  n 

tan  li  cos  (L  —  n  4-  h  n) 
cos  2  n 


(6) 


=  sin®- cos h  j^sin  (A  +  «r) 
(A  + 


cos 


Make  tan  y  = 


_  tan  /i  cos  (L  —  n  4-  $  n) 


cos  2  n 


.  Then, 


sin  7r  =  sin  ®*  cos  /i  sin  (A  4-  —  _in  cos  (A  +  I 

L  cos  y  J 

30S  hr-,  -  ,  _  .  "I 

— —  |^sin  (A  +  *■)  cos  y  —  cos  (a  4-  *)  sm  y J 
sin  (a  -f  »  —  y)  (H) 


sin  -sr  cos  h 
cos 
sin  ®*  cos  h 


cos  y 

sin  w  cos  h  .  ,  .  -v 

sm  (A  —  1/)  cos  *■ 


cos  y 


cos  (A  —  y)  sin 


1  = 


cos  y 

sin  w  cos  h  .  .  .  ,  sin®- cos /i  ,  . 

sm  (A  — y)  cot  5r  H - cos  (A — y ) 


cos  y 


COt  IF  = 


cos  y 


sin  ®-  cos  h  sin  (A  —  y 


cos  y 
cot  (A  —  y ) 


Make  tan  v  = 


sin  ®-  cos  h  sin  (A  —  y) 
cos  y 


a) 


Then  cot  «•  =  cot  v  —  cot  (A  —  y)  =  - ^ - — 

sin  v  sin  (A  — y) 

(App.  21) 

TT  .  sin  (A  —  y)  sin  v 

sin  (A  —  y  —  v) 

16 


114 


astronomy. 


EFFECT  OF  PARALLAX  ON  THE  APPARENT  DIAMETER 
OF  THE  MOON. 


57.  The  moon  is  nearer  to  any  place  on  the  earth’s 
surface  when  it  is  elevated  above  the  horizon,  than 
when  it  is  in  the  horizon.  The  angle  under  which  its 
diameter  is  seen,  will  therefore  be  greater  in  the  for¬ 
mer  case,  than  in  the  latter. 


Let  S'  =  moon’s  horizontal  semidiameter, 

S''  —  moon’s  apparent  semidiameter  at  a  given  situation 
above  the  horizon.  Then  (7.13), 

sin  S'  __  D  .p."  gx  BC  _  sinBAZ  __  sin  (N  +  p)  „ 

■sinT  =  D7  =  C  lg'  j  AB  “  sTnBCZ  ii^N  C  ’  '' 
_  sin  (A  +  g-)  sin  f  —  it  +  n) 

sin  A  sin  (  L  —  n) 
sin  ^  sin  (A  4-  *■)  sin  (L  —  n  4-  n) 


Hence  sin  S'  = 


sin  A  sin  (L  —  n ) 


(D 


We  may  obtain  other  formulae  for  expressing  the  relation  be¬ 
tween  S  and  which  will  be  more  convenient  when  the  paral¬ 
lax  in  latitude  is  found  by  the  second  method. 

Let  Ps  r,  Fig.  25,  be  the  arc  of  a  great  circle,  bisecting  the 
angle  ApB,  and  ZGsL,  another  arc  of  a  great  circle,  perpen¬ 
dicular  to  pr.  Then  pG  —  pL,  the  angle  pGL  ==  pLG,  Gps  ==  § 
n,  and  Zps  =  npr  =  npa  +  apr  =  L  —  n  4-  |n. 


Hence  tan  ps  ==  tan  pZ  cos  Zps  =  tan  h  cos  (L  —  n  +  5  n) 

~  tan  ps  tan  h  cos  (L  —  n  +  h  n) 

tan  pG  = - - —  =  - v — - - =  tan  y 

cos  Gps  cos  5  n 

(56) 


Therefore  pG  =  p,  AG  =  A p  —  Gp  =  A  —  y  and  BL  = 
Bp  —  Lp  =  A  -f  *•  —  y. 


Now 

Hence 

sin  BL  sinBZL 

sin  AZG 

sin  AG 

sin  BZ  sin  L 

sin  BL  sin  BZ 
sin  AG  sin  AZ 

sin  G 

sin  AZ* 

or 

sin  (A  +  ?r  —  y 

sin  (N  +  p) 

sin  S' 

sin  (A  —  y) 

sin  N 

sin  S 

CHAPTER  XI. 


115 


and  sin 


(M) 


sin  £  sin  (A  -f  57  —  y ) 
sin  (A  y) 

The  last  formula  may  be  reduced  to  another  still  more  simple 
For  we  have  (56  H  and  I), 

•  *.  >  sin  w  cos  w  .  .  , .  .  tan  v  cos  y 

sm  (A  -f  n  —  in  =  — - f.and  sin  (A  —  y)  —  — - p 

V  1  civ.  nr\c  7i*  ^  ^  '  Sin  ZT  COS  ll 


sinar  cos  h 

TT  sin  (A  +  7r  —  ?/)  sin  «■ 

Hence, - - -  =  — — , 

sin  (A  —  y)  tan  v 

sin  ^sin  *■ 


and  sin  = 


tan  v 


(N). 


CHAPTER  XI. 

Eclipses  of  the  Sun  and  Moon. — Occultations . 

1.  As  the  moon  is  an  opaque  body,  and  shines,  only, 
by  reflecting  the  sun’s  light,  when,  at  the  time  of  full 
moon,  it  enters  the  earth’s  shadow,  it  must  become 
eclipsed.  When,  at  the  time  of  new  moon,  the  moon 
passes  between  the  sun  and  a  spectator  on  the  earth,  it 
must  occasion,  to  him,  an  eclipse  of  the  sun. 

2.  If  we  suppose  the  sun  and  earth  to  be  spheres, 
as  they  are,  very  nearly,  the  sun  being  much  larger 
than  the  earth,  the  shadow  of  the  earth  must  have  the 
form  of  a  cone,  the  length  of  which  depends  on  the 
magnitudes  of  the  bodies,  and  on  their  distance  from 
each  other.  The  moon’s  shadow  is  also  conical,  but  of 
less  extent  than  that  of  the  earth. 

earth’s  shadow. 

3.  Let  ABG  and  abg ,  Fig.  26,  be  sections  of  the  sun 
and  earth,  by  a  plane,  passing  through  their  centres 
S  and  E,  and  AaC  and  B6C,  tangents  to  the  circles 
ABG  and  abg.  Disregarding,  at  present,  the  action  of 


116 


ASTRONOMY. 


the  earth’s  atmosphere,  in  changing  the  direction  of 
those  rays  of  the  sun  which  pass  through  it,  the  tri¬ 
angular  space  aCb  will  be  a  section  of  the  earth’s 
shadow. 

The  line  EC  is  called  the  Axis  of  the  earth’s 
shadow. 

4.  With  a  view  to  conciseness  of  expression  in  some 
of  the  succeeding  articles,  we  shall  put  R  =  E b  =  ra¬ 
dius  of  the  earth,  P  =  moon’s  horizontal  parallax, 
J)  =  sun’s  horizontal  parallax,  d  =  moon’s  apparent 
semidiameter,  and  <1  =  sun’s  apparent  semidiameter. 

5.  The  earth’s  shadow  extends  to  more  than  twice 
the  distance  of  the  moon. 

Put  n  =  206264". 8  (5.8). 

From  the  triangle  EBC,  we  have, 

SEB  =  EC6  +  EB6,  or  *  =  EC b  +  p. 

Therefore,  EC5  =  S  —  p, 

,  T1~  R  R .ft  R.n 

and  EC  =  .  _^f  =  =  - - . 

sinEC6  EC  b  S — p 

R<  )i 

Now  (5.8),  the  moon’s  distance  from  the  earth  =  1-. 

Hence,  as  S  —  p  is  less  than  half  P,  the  distance  EC,  is  more 
than  double  the  moon’s  distance  from  the  earth. 

6.  Let  hMh'  be  a  circular  arc,  described  with  the 
centre  E,  and  a  radius  equal  to  the  distance  between 
the  centres  of  the  earth  and  moon;  and  let  A dh  and 
B ch'  be  tangents  to  the  sections  of  the  sun  and  earth, 
crossing  each  other  between  them. 

When  any  part  of  the  moon  enters  the  space  be¬ 
tween  the  lines  dh  and  5C,  that  part  will,  evidently, 
be  deprived  of  a  portion  of  the  sun’s  light,  and  will 
therefore  appear  less  bright.  As  the  moon  approaches 


CHAPTER  XI. 


117 


the  line  bG,  its  light  continues  to  be  diminished;  and 
and  when  the  edge  comes  in  contact  with  bC,  the 
eclipse  commences.  Hence  there  is  a  gradual  diminu¬ 
tion  of  the  moon’s  light,  previous  to  the  commencement 
of  an  eclipse  of  the  moon.  There  is  also  a  gradual  in¬ 
crease  in  the  light,  after  the  eclipse  has  ended.  This  is 
conformable  to  observation. 

7-  If  we  suppose  the  line  dh  to  revolve  about  EC, 
and  form  the  surface  of  the  frustum  of  a  cone,  of  which 
cdlih'  is  a  section,  the  space  included  within  that  sur¬ 
face  is  called  the  Penumbra.  The  earth’s  shadow  is 
sometimes  called  the  Umbra. 

8.  The  moon  sometimes  enters  the  penumbra,  and 
again  passes  out,  without  any  part  entering  the  umbra 
or  real  shadow.  In  such  cases,  it  sustains  a  diminu¬ 
tion  of  its  light,  but  is  not  said  to  be  eclipsed. 

9.  Any  section  of  the  earth’s  shadow  or  of  the  pe¬ 
numbra,  by  a  plane  perpendicular  to  the  axis  of  the 
shadow,  is  a  circle.  If  we  suppose  such  a  section  of 
the  shadow  and  penumbra,  to  be  made  at  the  distance 
of  the  moon,  the  apparent  semi-diameter  of  the  section 
of  the  earth’s  shadow,  as  seen  from  the  centre  of  the 
earth,  is  called  the  Semi-diameter  of  the  Earth’s  Sha¬ 
dow;  and  the  apparent  semi-diameter  of  the  section  of 
the  penumbra,  is  called  the  Semi-diameter  of  the  Pe¬ 
numbra. 

The  angle  MEm  is  the  semi-diameter  of  the  earth’s 
shadow,  and  the  angle  MEZi'  is  the  semi-diameter  of 
the  penumbra. 

10.  The  semi-diameter  of  the  earth’s  shadow  is 
ecpial  to  the  sum  of  the  moon  and  sun’s  horizontal  pa¬ 
rallaxes,  less  the  apparent  semi-diameter  of  the  sun. 

Tn  the  triangle  EmC,  we  have, 

the  angle  mEM  =  E mb  —  ECm. 


118 


ASTRONOMY. 


Now,  the  angle  mEM  is  the  semi-diameter  of  the  earth’s  sha¬ 
dow,  E mb  is  the  moon’s  horizontal  parallax  =  P,  and  ECm  = 

P  c5)- 

Hence, 

Semi-diameter  of  the  Earth’s  Shadow  =  P  —  (<^  —  p)  =  P  4- 
p-*- 

11.  The  diameter  of  the  earth’s  shadow,  at  the 
moon,  is  more  than  double  the  apparent  diameter  of  the 
moon,  and  consequently  the  moon  may  be  entirely  en¬ 
veloped  in  the  shadow. 

If  we  take  P  =  57'  22",  p  —  8". 7,  and  ^  =  16'  1".3,  we 
obtain  the  mean  semi-diameter  of  the  earth’s  shadow  =  41' 
29". 4,  and  consequently  the  mean  diameter  =  82'  58". 8;  which 
is  more  than  twice  the  apparent  diameter  of  the  moon. 

12.  If,  at  the  time  of  full  moon,  the  apparent  dis¬ 
tance  of  the  moon’s  centre,  from  the  axis  of  the  shadow, 
does  not  become  less  than  P  4.  y  -f  d  —  <T,  there  can 
not  be  an  eclipse. 

When  the  edge  of  the  moon  touches  the  earth’s  shadow;  its 
centre  is  at  a  distance  from  it,  equal  to  the  moon’s  semi-diameter. 
If,  therefore,  to  the  semi-diameter  of  the  shadow,  we  add  the 
moon’s  semi-diameter,  we  have  the  distance  of  the  moon’s  centre 
from  the  axis  of  the  shadow,  at  the  beginning  or  end  of  an  eclipse 
of  the  moon,  equal  P  -f  p  +  d  — 

43.  The  distance  of  the  moon’s  centre  from  the  axis 
of  the  shadow,  at  the  time  of  full  moon,  depends  on  the 
moon’s  distance  from  the  node,  and  on  the  inclination 
of  the  orbit.  W e  may,  for  a  given  inclination  of  the 
orbit,  and  given  value  of  P  4-  p  4-  d  —  <1,  determine 
within  what  distance  from  the  node,  the  moon  must  be, 
in  order  that  an  eclipse  may  take  place. 


CHAPTER  XI. 


149 


By  taking  the  least  and  greatest  inclinations  of  the 
orbit,  the  greatest  and  least  values  of  P  4-  p  +  d  — 
and  also  taking  into  view  the  inequalities  in  the  mo¬ 
tions  of  the  sun  and  moon,  it  has  been  found,  accord¬ 
ing  to  Delambre,  that  when  at  the  time  of  mean  full 
moon,  the  difference  of  the  mean  longitudes  of  the 
moon  and  node,  exceeds  12°  38',  there  can  not  be  an 
eclipse;  but  when  this  difference  is  less  than  9°  there 
must  be  one.  These  numbers  are  called  the  Lunar 
Ecliptic  Limits. 

14.  From  tables  of  the  mean  motions  of  the  sun, 
moon,  and  node,  the  mean  time  of  any  full  moon,  and 
the  difference  of  the  mean  longitudes  of  the  moon  and 
node,  at  that  time,  are  easily  obtained.  Then  by  the 
lunar  ecliptic  limits,  we  know  whether  or  not  these  will 
be  an  ecliptic,  except  when  the  difference  of  longitudes 
of  the  moon  and  node,  is  between  9°  and  12°  36',  in 
which  case  further  calculation  is  necessary. 

15.  W e  have  hitherto,  considered  the  earth’s  shadow 
as  limited  by  those  rays,  which,  passing  from  the  edge 
of  the  sun,  touch  the  corresponding  side  of  the  earth. 
But  it  is  found  that  the  observed  duration  of  an  eclipse 
of  the  moon,  always  exceeds  the  duration,  computed 
on  the  supposition  of  the  shadow  being  thus  limited. 
This  is  accounted  for,  by  supposing  the  most  of  those 
rays  of  light  which  pass  near  the  surface  of  the  earth 
to  be  absorbed  by  the  earth’s  atmosphere,  as  the  effect 
of  such  an  absorption  would  evidently  b  an  increase 
in  the  extent  of  the  earth’s  shadow.  In  consequence 
of  the  gradual  diminution  of  the  moon’s  light,  previous 
to  its  entering  the  earth’s  shadow,  and  gradual  increase 
on  leaving  it  (6),  the  time  of  the  commencement  or  end 
of  a  lunar  eclipse  can  not  be  observed  with  great  accu¬ 
racy.  Astronomers  therefore  differ  with  regard  to  the 


120 


ASTRONOMY. 


amount  of  the  correction  which  should  be  made.  It 
is  however  usual  in  computing  an  eclipse  of  the  moon, 
to  increase  the  semi-diameter  of  the  earth’s  shadow  by 
a  ^  part;  or  which  amounts  to  the  same,  to  add  as 
many  seconds  as  the  semi-diameter  contains  minutes. 

16.  There  is  another  effect  of  the  earth’s  atmosphere, 
which  is  perceptible  in  eclipses  of  the  moon.  Those 
rays  of  the  sun  which  enter  the  earth’s  atmosphere  and 
are  not  absorbed,  have  their  directions  changed,  so  as 
to  meet  at  a  less  distance  than  they  otherwise  would 
do.  The  rays  through  the  atmosphere  near  the  sur¬ 
face  of  the  earth,  are  so  much  converged  as  to  meet  at 
a  distance  from  the  earth,  less  than  that  of  the  moon. 
In  this  way  a  sufficient  quantity  of  light  is  thrown  on 
the  moon  to  render  it  visible,  even  when  it  is  in  the 
middle  of  the  shadow.  It  then  appears  with  a  dull,  red¬ 
dish  light. 

17.  As  an  eclipse  of  the  moon  is  occasioned  by  a 
real  loss  of  light  on  the  moon,  and  not  by  the  interpo¬ 
sition  of  any  body  between  the  moon  and  the  spectator 
on  the  earth,  it  must  present  the  same  appearance  to 
all  those  who  have  the  moon  above  their  horizon  during 
the  eclipse,  and  observe  it  at  the  same  time.  It  will 
be  shown  that  the  case  is  different  with  eclipses  of  the 
sun. 

18.  The  semi-diameter  of  the  earth’s  penumbra  is 
equal  to  the  sum  of  the  moon’s  horizontal  parallax,  sun’s 
horizontal  parallax,  and  the  apparent  semidiameter  of 
the  sun. 

From  the  triangles  EL/t',  and  ELB,  we  have, 

MEh'  =  E/i'c  4-  ELc  =  E lie  4-  EBc  4-  SEB. 

But  E/i'c  =r  P,  EBc  p,  and  SEB  =  Hence, 
Semi-diameter  of  the  Penumbra  =  P  T  p  -f  £ 


CHAPTER  XI. 


121 


moon’s  shadow, 


19.  The  length  of  the  moon’s  shadow  is  about  equal 
to  the  distance  of  the  moon  from  the  earth,  being,  some¬ 
times,  a  little  greater,  and,  sometimes,  a  little  less. 

Let  now  agb  be  considered  as  a  section  of  the  moon,  and  M 
the  centre  of  the  earth,  then,  the  same  figure  will  answer  for  de¬ 
termining  the  different  circumstances  relative  to  the  moon’s  sha¬ 
dow,  that  have  been  found  for  the  earth’s  shadow.  Put, 

R'  =  E6  =  moon’s  radius, 

p '  =  FiBb  —  sun’s  horizontal  parallax  for  the  moon, 


—  SEB  =  sun’s  apparent  semi-diameter  at  the  moon. 
We  have,  (5.8) 

SM  =  —y  and  EM  =  5^. 


P 

Now,  ES  =  SM  —  EM  == 

P 


P  * 

R.n  _  R.n(P — p) 
“P  “  P 7P 


Hence,  (7.13)  = 


ES  R ,n  (P—  p)  P— p 


P 


P.^ 


We  have  also  (7.15)  R'  =  —.and SB  =  —  =  — . 

P  P  P' 


Hence,  p' 


p.R'.'P  _  p  R A  P.^  _  p.d 
R.^  R.J1  ‘  P  ’  P  —  p  ~  P  —  p 


ASTRONOMY, 


122 


R.u 


In  the  last  expression  for  the  value  of  EC,  the  factor  ex¬ 
presses  the  moon’s  distance  from  the  centre  of  the  earth.  And  it 

p -p 

is  evident  the  other  factor  <5^ _  is  greater  than,  equal  to,  or 

d  V 


less  than  a  unit,  according  as  d  is  greater  than,  equal  to,  or  less 
than  £  Hence  the  moon’s  shadow  extends  beyond  the  centre  of 
the  earth,  just  to  it,  or  not  so  far,  according  as  the  moon’s  apparent 
diameter  is  greater  than,  equal  to,  or  less  than  the  sun’s  apparent 
diameter. 

20.  The  apparent  semi-diameter,  as  seen  from  the 
moon,  of  a  section  of  the  moon’s  shadow,  at  the  earth, 
is  equal  to, 

<'  -'>•  p-h 


From  the  triangle  mEC,  we  have, 
mEM  =  E  mb  —  ECm  =  d 
p.d  P  ^ 


—  (19),  d  -j- 


P 


P -P 


—p')  =  d  +p'  — 

P  —P 


S' 


P 

P  —  P 


21.  The  greatest  breadth  of  the  moon’s  shadow,  at 
the  earth,  is  about  TXT.  part  of  the  earth’s  diameter.  The. 
expression  for  the  breadth  is, 


d—J 
V — p 


,  2U. 


To  obtain  the  breadth,  we  have 
Mm  *=  ang*e-MEm  __  l  .  R.n 


(d— <F).  P 
P  -P 


CHAPTER  XI. 


128 


or  mm'  =  2  Mm  =  2  R, 

If  we  take  1 1  =  I005".5,  S  =  945".5,  P  =  3692",  and  p 
•=  8". 7,  which  are,  the  greatest  value  of  d,  least  of  and  the 
corresponding  values  of  P  and  p,  we  evidently  obtain  the  greatest 
value  of  mm'.  These  numbers  give, 


mm’  =  F=f  2  R  =  368T3-  2  R  =  tTf  2  R  near,y- 


22.  As  the  sun  can  only  be  entirely  eclipsed  at  those 
parts  of  the  earth’s  surface  on  which  the  moon’s  sha¬ 
dow  falls,  it  is  evident,  from  the  preceding  article,  that 
in  the  most  favourable  cases,  this  phenomenon  can  only 
have  place  for  a  small  portion  of  the  earth. 

23.  When  for  a  point  on  the  earth,  in  the  right  line 
passing  through  the  centres  of  the  sun  and  moon,  d  = 
the  breadth  of  the  shadow  will  be  nothing.  In  this 
case,  the  sun  will  be  entirely  eclipsed,  but  it  will  not 
continue  so,  for  any  perceptible  time.  When  d  is  less 
than  the  expression  for  the  breadth  of  the  shadow 
becomes  negative;  the  rays  from  the  edge  of  the  sun, 
which  pass  near  the  moon,  crossing  each  other,  in  that 
case,  before  they  arrive  at  the  earth.  In  those  parts  of 
the  earth  where  this  has  place,  the  edge  of  the  sun  will 
appear  as  a  ring,  surrounding  the  moon. 

24.  The  apparent  semi-diameter,  as  seen  from  the 
moon,  of  a  section  of  the  moon’s  penumbra,  at  the  earth, 
is  equal  to 


( d  4*  <1). 


From  the  triangles  EL/i'  and  ELB,  we  have, 

ME/i'  =  E/i'c  4-  ELc  =  Eh'c  4-  EBc  4-  SEB  =  d  4-  p’  4-  2- ' 


.=  d  4- 


p.d  VJ 
P -p  +  p  — j 


P.d  4-  P.^ 


-  C  +  '>•  p4? 


124 


ASTRONOMY. 


25.  The  greatest  breadth  of  the  moon's  penumbra, 
at  the  earth,  is  a  little  more  than  half  the  earth’s  dia¬ 
meter.  The  expression  for  the  breadth  is, 


d  +  cT 
P  —  p 


2R. 


To  obtain  the  breadth,  we  have, 

, ,  __  EM.  ang.  ME/i'  _  1  R.?z  (d  -f  F).  P 
~  n  n  T"* 

and  hh'  =  2  MV  =  d  ~t-.  2  R. 

F—p 


d+t 

P  ~P 


R. 


Taking  d  =  881",  ?  =  977."8,  P  =  3232"  and  p  =  8".7, 
we  obtain  she  greatest  value  of, 


hh'  =  2  R  =  1—. 

P  —  p  3223 


2  R  =  2  R  nearly. 

40  J 


26.  As  no  part  of  the  sun  can  be  hid  by  the  moon, 
at  those  parts  of  the  earth  which  are  without  the  pe¬ 
numbra,  the  sun  may  be  wholly  visible  for  a  large  por¬ 
tion  of  the  earth,  ,while  it  is  eclipsed  either  in  part  or 
entirely,  in  other  parts. 

27.  If,  at  the  time  of  new  moon,  the  apparent  dis¬ 
tance  of  the  sun  and  moon  does  not  become  less  than 
P  —  p  +  d  +  there  can  not  be  an  eclipse  of  the 
sun  to  any  part  of  the  earth. 

Again,  considering  agb  as  a  section  of  the  earth,  let  M'  be  the 
place  of  the  moon’s  centre,  when  in  the  conical  surface,  which 
circumscribes  the  sun  and  earth.  Then  the  angle, 

M'ES  =  EM  6  +  EC6  =  P  —  p  +  * 

If  to  the  value  of  M'ES,  we  add  the  apparent  semi-diameter 
of  the  moon,  we  shall  have,  for  the  apparent  distance  of  the  cen¬ 
tres  of  the  sun  and  moon,  at  the  beginning  or  end  of  an  eclipse  of 
the  sun,  the  expression  P — p  4*  d  -f 


CHAPTER^!. 


125 


28.  From  the  expression  P  —  p  4-  d  +  ^  by  taking 
into  view  the  inclination  of  the  orbit,  and  the  inequali¬ 
ties  in  the  motions  of  the  sun  and  moon,  it  has  been 
found,  according  to  Delambre,  that  when  at  the  time  of 
mean  new  moon,  the  difference  of  the  mean  longitudes 
of  the  moon  and  node  exceeds  19°  2',  there  can  not  be 
an  eclipse  of  the  sun:  but  when  this  difference  is  less 
than  13°  14/,  there  must  be  one.  These  numbers  are 
called  the  Solar  Ecliptic  Limits . 

29.  As  the  solar  ecliptic  limits  exceed  the  lunar, 
eclipses  of  the  sun  must  occur  more  frequently  than 
those  of  the  moon.  But  as  the  former  are  only  visible 
to  some  parts  of  that  portion  of  the  earth,  which  has 
the  sun  above  the  horizon  during  the  eclipse  (26),  and 
the  latter  to  the  whole  of  that  portion  which  has  the 
moon  above  the  horizon  (17),  there  are,  for  any  given 
place,  more  visible  eclipses  of  the  moon  than  of  the 
sun. 

NUMBER  OF  ECLIPSES  IN  A  YEAR. 

30.  From  the  solar  and  lunar  ecliptic  limits,  and  the 
motions  of  the  sun,  moon,  and  node,  it  is  found  that 
the  greatest  number  of  eclipses  that  can  take  place  in 
a  year  is  seven;  and  that  the  least  number  is  two. 

When  there  are  seven  eclipses  in  a  year,  five  are  of 
the  sun,  and  two  of  the  moon.  When  there  are  only 
two,  they  are  both  of  the  sun.  In  every  year  there  are 
at  least  two  eclipses  of  the  sun. 

DIFFERENT  KINDS  OF  ECLIPSES. 

31.  When  the  moon  just  touches  the  earth’s  shadow, 
or  approaches  very  near,  without  entering  it,  the  cir¬ 
cumstance  is  called  an  Appulse .  When  a  part,  but  not 


126 


ASTRONOMY. 


tlie  whole  of  the  moon,  enters  the  earth’s  shadow,  the 
the  phenomenon  is  called  a  Partial  eclipse  of  the  moon; 
when  the  moon  enters  wholly  into  the  shadow,  it  is 
called  a  Total  eclipse;  and  when  the  centre  of  the  moon 
passes  through  the  axis  of  the  shadow,  the  eclipse  is 
said  to  be  Central.  An  exactly  central  eclipse  of  the 
moon  seldom,  if  ever  occurs. 

With  regard  to  the  sun,  when  the  disc  of  the  moon 
just  touches,  or  approaches  very  near,  to  the  disc  of  the 
sun,  the  circumstance  is  called  an  Appulse.  When  the 
moon  obscures  a  part,  and  only  a  part,  of  the  sun,  the 
eclipse  is  said  to  be  Partial;  and  when  the  moon  oh- 
scures  the  whole  of  the  sun,  the  eclipse  is  said  to  be 
Total.  When  the  moon’s  disc  is  entirely  interposed 
between  the  spectator  and  the  sun,  but  in  consequence 
of  the  apparent  diameter  of  the  moon,  being  less  than 
that  of  the  sun,  the  edge  of  the  sun  is  seen  as  a  ring 
surrounding  the  moon  (23),  the  eclipse  is  called  An¬ 
nular.  Lastly,  when  the  straight  line  passing  through 
the  centres  of  the  sun  and  moon,  passes  also  through 
the  place  of  the  spectator,  the  eclipse  is  said  to  be  Cen¬ 
tral. 

ECLIPSES  OF  THE  MOON. 

32.  The  apparent  distance  of  the  centre  of  the  moon 
from  the  axis  of  the  earth’s  shadow,  and  the  arcs  of  the 
moon’s  orbit  and  of  the  ecliptic  passed  through  by  these, 
during  an  eclipse  of  the  moon,  being  necessarily  small, 
may  without  material  error,  be  considered  as  right  lines. 
We  may  also  consider  the  apparent  motion  of  the  sun 
in  longitude  and  the  motions  of  the  moon,  in  longitude 
and  latitude,  as  uniform,  during  the  eclipse.  These 
suppositions  being  made  the  calculation  of  the  circum¬ 
stances  of  an  eclipse  of  the  moon,  is  very  simple. 


CHAPTER  XI. 


33.  Let  NF,  Fig.  27,  be  a  part  of  the  ecliptic,  NL  a  part  of  the 
moon’s  orbit,  C  the  centre  of  a  section  of  the  earth’s  shadow  at 
the  moon,  CD  perpendicular  to  NF,  a  circle  of  latitude,  and  M 
the  centre  of  the  moon  at  the  instant  of  opposition.  Then  CM,, 
which  is  latitude  of  the  moon, In  opposition,  is  the  distance  of  the 
centres  of  the  shadow  and  moon  at  that  time. 

Let  t  be  some  short  interval  of  time  expressed  in  hours,  and 
parts  of  an  hour,  and  let  C'  and  M'  be  the  situation  of  centres 
of  the  shadow  and  moon  at  the  time  t  before  or  after  opposition. 
Then  CM'  will  be  the  distance  of  the  centres  at  that  time.  Draw 
M'F  perpendicular,  and  AMB  parallel  to  NF.  Then  CC'  is  the 
motion  of  the  centre  of  the  shadow  in  the  time  f,  CF  is  the  moon’s 
motion  in  longitude,  and  HM'  its  motion  in  latitude.  Now  as  the 
longitude  of  the  centre  of  the  earth’s  shadow,  must  always  differ 
by  180°,  from  the  longitude  of  the  sun,  the  apparent  motion  of  the 
sun  is  the  same  as  that  of  the  centre  of  the  shadow.  Therefore 
CC'  expresses  the  sun’s  motion  in  longitude  in  the  time  t .  And 
consequently  C'F  =  CF  —  CC'  =  the  difference  of  the  moon’s 
and  sun’s  motions  in  longitude,  in  the  time  t. 

34.  Make  CG  equal  to  C'F,  and  GM"  perpendicular  to  NF,  and 
equal  to  FM'.  Then  CM"  =  C'M'  =  the  distance  of  the  centres 
of  the  moon,  and  earth’s  shadow  at  the  time  f,  from  opposition. 
We  therefore  obtain  the  distance  of  the  centres  of  the  moon,  and 
shadow,  the  same,  if  instead  of  allowing  to  each  its  proper  motion 
we  suppose  the  centre  of  the  shadow  to  remain  at  rest  at  C,  and 
the  moon’s  motion  in  longitude  to  be  equal  to  the  difference  of  the 
motions  of  the  moon  and  sun,  in  longitude. 

35.  From  Astronomical  Tables  we  can  get  the  hourly  motions  of 
the  sun  and  moon,  in  longitude,  and  the  moon’s  hourly  motion  in 
latitude.  Then  supposing  the  motions  uniform,  we  easily  obtain 
their  values  for  any  other  short  interval  of  time.  Put 

T  =  time  of  opposition, 

t  =  time  of  moon’s  centre  passing  from  Mto  M', 
m  —  moon’s  hourly  motion  in  longitude, 
n  =  moon’s  hourly  motion  in  latitude, 
r  sun’s  hourly  motion  in  longitude, 

*  =  moon’s  latitude  at  opposition, 


128 


ASTRONOMY. 


I  *=  angle  M"MR, 

s  =  P  -f  ;>  —  ^  +  -tv  (P  4-  p  —  ^)  =  semidiam.  of 
earth’s  shadow  (10  and  15). 


Then  CF  =  m.f,  CC'  =  r.t  and  RM"  =  HM'  =  n.t, 

MR  =  CG  =  C  F  =  CF —  CC'  =  m.t  — r.t  —  t.  (m  —  r)y 


tan.  I  =  tan  M  "MR  = 


RM''  __  t.n 
RM  t.  (m — r) 


As  the  expression  for  the  tangent  of  the  angle  M"MR  does  not 
involve  f,  it  is  evident  the  angle  itself  will  continue  the  same, 
whatever  be  the  value  of  t.  Hence  the  point  M"  moves  in  the 
line  PMQ,  which  is  therefore  called  the  moon's  Relative  Orbit . 

36.  In  the  triangle  M'MR,  we  have, 


MM"  = 


MR 

cos  M'MR 


t.(m  —  r) 
cos  I 


The  distance  MM"  is  the  moon’s  motion  on  the  relative  orbit, 
in  the  time  t.  If  we  take  t  —  1  hour,  we  have, 

7H  T 

The  moon’s  hourly  motion  on  relative  orbit  =  - - . 

cos  I 

37.  Let  AB,  Fig.  28,  be  the  ecliptic,  C  the  centre  of  the  earth’s 
shadow  at  the  time  of  opposition,  and  CK  perpendicular  to  AB,  a 
circle  of  latitude.  Make  CM  =  a,  Mb  parallel  to  AB,  and  =  m 
—  r,  and  be  parallel  to  CK,  and  =  n.  Through  M  and  c,  draw 
DMcH,  which  will  be  the  moon’s  relative  orbit.  With  the  centre 
C  and  a  radius  =  s,  describe  the  circle  KLPI,  which  will  repre¬ 
sent  the  section  of  the  earth’s  shadow  at  the  moon.  With  the  same 
centre  and  a  radius  =  s  -f  d,  describe  arcs  cutting  DH  in  D  and 
H;  and  with  a  radius  —  s  —  d,  describe  other  arcs,  cutting  DH 
in  E  and  G.  From  C,  draw  CF  perpendicular  to  DH.  Then 
supposing  the  moon  to  move  in  the  direction  DII,  it  is  evident  that 
D  is  the  place  of  the  moon’s  centre  at  the  beginning  of  the  eclipse; 
E,  its  place  at  the  beginning  of  the  total  eclipse;  F,  its  place,  when 
nearest  the  centre  of  the  shadow;  G,  its  place  at  the  end  of  the  to¬ 
tal  eclipse;  and  H,  its  place  at  the  end  of  the  eclipse.  When 
s  —  cl  is  less  than  CF,  the  eclipse  can  not  be  total. 

38.  Because  CD  =  CH,  and  CF  is  perpendicular  to  DH,  we 


CHAPTER  XI. 


129 


have  FD  =  FH.  The  point  F,  therefore  designates  the  moon’s 
place  at  the  middle  of  the  eclipse. 

In  the  triangles  MFC  and  Mbc,  the  angles  F  and  b  are  right 
angles;  and  because  be  is  parallel  to  CM,  the  angle  FMC  =  Mc&. 
Therefore,  MCF  =*=  bMc  =  I  (35). 

MIDDLE  OF  THE  ECLIPSE. 

39.  In  the  triangle  MCF, 

MF  =  CM  sin  MCF  =  a  sin  I. 

But,  taking  x  =  interval  of  time  between  the  middle  $f  the 
eclipse,  and  the  time  of  opposition,  we  have  (36), 


MF  =  *•<”— r). 

COS  I 


Hence, 


x.  (m — r) 


a  sin  I,  or  x  = 


a  sin  I  cos  I 


cos  I  m  —  t 

Now  if  M  =  the  time  of  the  middle,  we  obtain, 

a  cos  I  sin  I 


M  =  T  ?  i  =  T  ? 


m 


The  upper  sign  must  be  used  when  the  latitude  is  increasing ; 
and  the  lower ,  when  it  is  decreasing. 

The  nearest  distance  of  the  centre  is  CF  =  a  cos  I. 


BEGINNING  AND  END  OF  THE  ECLIPSE. 

40.  Let  B  =  the  time  of  beginning,  E  =  the  time  of  the  end, 
and  x  =  the  interval  between  the  middle  and  either  of  these. 
Then, 

=  DF  = 


FC 


cos 


=  >/  (s  -f-  d)2  —  a2  cos2  I 


=*  */  (s  4-  d  —  a  cos  I).  (s  +  d  -f  a  cos  I) 

qt  x  —  cos  I  >/  (g  4-  d  —  a  cos  I  .  (s  4.  d  +  a  cos  1) 

m  —  r 

Hence,  B  =  M  — -  rr,  and  E  *=  M  -f  a,  become  known, 
48 


130 


ASTRONOMY. 


BEGINNING  AND  END  OF  THE  TOTAL  ECLIPSE. 

41.  Put  B'  =  the  time  of  the  beginning  of  the  total  eclipse, 
E'  =  the  time  of  the  end,  and  x'  =  the  interval  between  each 
of  these  and  the  middle.  Then, 

__  cos  I  V  (s —  d —  A  cos  I).  (5 —  d  4-  cos  I) 
m  —  r 

B'  =  M  —  x \  and  E'  =  M  4-  x'. 

QUANTITY  OF  THE  ECLIPSE. 

42.  In  an  eclipse  of  the  moon,  it  is  usual  to  suppose  that  dia¬ 
meter  of  the  moon,  which,  produced  if  necessary,  passes  through 
the  centre  of  the  shadow,  to  be  divided  into  twelve  equal  parts, 
called  Digits ,  and  to  express  the  quantity  of  the  eclipse  by  the 
number  of  those  parts,  that  is  contained  within  the  shadow,  at  the 
time  when  the  centres  of  the  moon  and  shadow  are  nearest. 
When  the  moon  is  entirely  within  the  shadow  as  in  total  eclipses, 
the  quantity  of  the  eclipse  is  still  expressed  by  the  number  of 
digits  of  the  moon’s  diameter,  which  is  contained  in  that  part  of 
a  radius  of  the  shadow,  passing  through  the  moon’s  centre,  which 
is  intercepted  between  the  edge  of  the  shadow  and  the  inner  edge 
of  the  moon.  Thus,  the  number  of  digits  contained  in  SN,  ex¬ 
presses  the  quantity  of  the  eclipse,  represented  in  the  figure. 
Hence,  if  Q  =  the  quantity  of  the  eclipse,  we  have, 

0_  NS  _  CS  — CN_  CS  — (CF  — FN) 

'  TfNV  tVNV  “  t\NV 
CS  +  FN—  CF  __  12  (CS  +  FN  —  CF) 

“  ^NV  NV. 

12  (P  4 -p P+d A  cos  I) 

~2d 

_  (P  4-  p  4-  d —  ^ — acos  I).  6 
_ 

CONSTRUCTION  OF  AN  ECLIPSE  OF  THE  MOON* 

43.  The  times  of  the  different  circumstances  of  an  eclipse  of 
the  moon,  may  easily  be  determined  by  a  geometrical  construction. 


CHAPTER  IX. 


131 


within  a  minute  or  two  of  the  truth.  To  render  the  construction 
explicit,  suppose  the  time  of  opposition  to  be  8  h.  35  m.  20  sec. 
on  some  given  day.  Then,  as  60  minutes  :  35  m.  20  sec.  :  : 
moon’s  hourly  motion  on  relative  orbit  :  moon’s  distance  from  the 
point  M  at  8  o’clock.  If  this  distance  be  taken  in  the  dividers,  and 
laid  on  the  relative  orbit,  from  M  backwards  to  the  point  8,  it 
will  give  the  moon’s  place  at  that  hour.  Then  taking  in  the  di¬ 
viders,  the  moon’s  hourly  motion  on  the  relative  orbit,  and  laying 
it  on  the  orbit  from  8  to  9,  9  to  10,  10  to  11,  and  backwards, 
from  8  to  7,  and  7  to  6,  we  have  the  places  of  the  moon’s  centre 
at  those  hours  respectively.  By  dividing  the  hour  spaces  into 
quarters,  and  subdividing  these  into  5  minute  spaces  or  minute 
spaces,  we  easily  perceive  the  times  at  which  the  moon’s  centre 
is  at  the  points  D,  E,  F,  G,  and  II. 

ECLIPSES  OF  THE  SUN. 

44.  It  lias  been  shown  in  a  preceding  article  (&7), 
that  when  the  angular  distance  of  the  centres  of  the 
sun  and  moon  is  equal  to  P  —  jp  +  ^  4-  d,  the  edge  of 
the  moon  just  touches  the  luminous  frustum  of  a  cone, 
contained  between  the  sun  and  earth.  When  the  dis¬ 
tance  of  the  centres  becomes  less  than  P  — jp  +  ^4-  d, 
it  is  evident  the  moon  must  obscure  a  portion  of  the 
sun’s  light  to  some  part  of  the  earth’s  surface.  The 
instants  at  which  the  moon  commences  and  ceases  to 
prevent  any  part  of  the  sun’s  light  from  arriving  at  the 
earth,  are  called  the  Beginning  and  End  of  the  General 
Eclipse  of  the  sun.  These  times  may  be  obtained 
either  by  calculation  or  construction,  nearly  in  the  same 
manner  as  the  beginning  or  end  of  an  eclipse  of  the 
moon. 

45.  Although  the  calculation  of  the  general  eclipse 
of  the  sun  is  equally  simple  with  that  of  a  lunar  eclipse, 
it  is  very  different  when  the  object  is  to  determine  the 
circumstances  of  the  eclipse  for  any  particular  place. 


132 


ASTRONOMY. 


Then,  it  is  necessary  to  take  into  view  the  situation  of 
the  place  on  the  illuminated  surface  of  the  earth,  or, 
which  amounts  to  the  same,  to  consider  the  effects  of 
parallax.  This  circumstance  renders  the  calculation 
tedious,  at  least,  when  it  is  desired  to  give  to  the  re¬ 
sults,  all  the  accuracy  of  which  the  problem  is  sus¬ 
ceptible. 

46.  We  shall  first  give  a  method  of  obtaining  results  nearly  true, 
by  means  of  a  geometrical  construction.  When  the  construction  is 
carefully  performed,  on  a  large  scale,  the  error  in  the  time  of  be¬ 
ginning  or  end  will  not  exceed  one  or  two  minutes.  This  method 
will  therefore  suffice,  except  considerable  accuracy  is  required. 
We  shall  afterwards  give  a  method  of  obtaining  by  calculation, 
from  the  results  of  the  construction,  others  that  will  be  more  ac¬ 
curate. 

The  earth  will  still  be  considered  as  a  sphere,  and  accordingly 
the  reduced  latitude  of  the  place,  and  the  reduced  parallax  must 
be  used  (10.51). 

47.  A  section  of  the  earth  made  by  a  plane  passing  through  its 
centre,  perpendicular  to  the  line  joining  the  centres  of  the  earth 
and  sun,  is  called  the  Circle  of  Illumination.  It  forms  nearly  the 
boundary  between  the  enlightened  and  dark  parts  of  the  earth’s 
surface.  As  the  sun  is  larger  than  the  earth,  it  evidently  enlightens 
a  small  portion  more  than  one  half  of  the  earth.  The  enlightened 
part  is  still  further  increased  by  the  effect  of  the  earth’s  atmos¬ 
phere  in  refractiong  the  rays  of  light. 

48.  A  plane  between  the  earth  and  sun,  perpendicular  to  the 
straight  line  joining  their  centres,  and  at  a  distance  from  the  earth’s 
centre,  equal  to  the  distance  of  the  centres  of  the  earth  and  moon, 
is  called  the  Plane  of  Projection. 

49.  If  from  the  sun’s  centre  to  every  point  in  the  circumference 
of  the  circle  of  illumination,  straight  lines  be  conceived  to  be 
drawn,  they  will  form  the  surface  of  a  right  cone,  a  section  of 
which,  by  the  plane  of  projection  is  a  circle,  and  is  called  the 
Circle  of  Projection, 

50.  A  plane  passing  through  the  centre  of  the  sun  and  the  poles 


CHAPTER  XI. 


133 


■of  the  earth,  and  which  consequently  passes  through  the  earth’s 
centre,  is  called  the  Universal  Meridian,  When  by  the  diurnal 
rotation  of  the  earth  on  its  axis,  any  place  on  its  surface  is  brought 
to  coincide  with  this  plane,  the  sun  must  be  on  the  meridian  of  that 
place. 

51.  Let  S,  Fig.  29,  be  the  centre  of  the  sun,  E,  the  centre  of 
the  earth,  TSU  the  plane  of  the  universal  meridian,  and  let 
AUWT  and  PRQV,  each  conceived  to  be  perpendicular  to  the 
plane  of  the  paper,  be,  respectively,  the  circles  of  illumination  and 
projection.  If  D  be  a  place  on  the  earth’s  surface,  a  spectator  at 
D,  will  see  the  sun’s  centre  in  the  direction  of  the  line  DS,  which 
intersects  the  circle  of  projection  in  L.  The  point  L  is  called  the 
Projection  of  the  Sun's  Centre ,  for  the  spectator  at  D. 

52.  Let  DF  and  LMbe  each  perpendicular  to  TUS,  the  plane  of 
the  universal  meridian,  and  FG  and  MC  each  perpendicular  to 
ES,  the  line  joining  the  centres  of  the  earth  and  sun.  Then  DF  and 
LM  are  the  distances  of  the  points  D  and  Lfrom  the  universal  me¬ 
ridian;  and  FG  and  MC  are  the  distances  of  the  same  points  from 
a  plane  perpendicular  to  the  universal  meridian,  and  passing 
through  the  centres  of  the  sun  and  earth. 

53.  From  the  triangle  EQS,  we  have, 

CE$  =  EQU  —  ESU 

or  Apparent  semidiarneter  of  the  circle  of  Projection  =  P — p. 

54.  By  similar  triangles,  EU  :  CQ  :  :  ES  :  CS  :  :  EN  :  CM. 
But  CQ  =  EC  tan  CEQ  =  EC  tan  (P  —  p);  and  we  may  with¬ 
out  sensible  error  consider  EN  =  FG.  Hence, 

EU  :  EC  tan  (P  — p)  :  :  FG  :  CM  =  FG~  EC  tan(P—  p) 

EU 

Therefore,tanCEM=  SIM  —  FG  tanfP _ p) 

EC  EU  V  '  Fh 

or,CEM  =  §r 

FD 

In  like  manner,  MEL  =  — .  (P — p). 

EU 

If  X  =  MEL  ==  the  apparent  distance  of  the  projection  of  the 
sun’s  centre  from  the  universal  meridian,  Y  =  CEM  =  the  ap¬ 
parent  distance  of  the  projection  of  the  centre,  from  the  plane 


131 


ASTRONOMY. 


passing  through  the  centres  of  the  earth  and  sun,  perpendicular  to 
the  universal  meridian,  and  R  =  EU  ==  radius  of  the  earth,  we 
have, 


Y  FD 
R  * 


(P  —  p),  and  Y  = 


FG 
R  * 


(p  -p). 


55.  Let  AUBT,  Fig.  30,  be  a  section  of  the  earth  by  the  plane 
of  the  universal  meridian,  PP'  the  earth’s  axis,  P  the  north  pole, 
P'  the  south  pole,  TU  a  diameter  of  the  circle  of  illumination,  CQ 
a  diameter  of  the  equator,  ES  the  line  joining  the  centres  of  the 
earth  and  sun,  and  D  a  given  place  on  the  earth.  Also  let  DLH 
be  a  plane  perpendicular  to  PP'  the  earth’s  axis,  DF  a  straight 
line  perpendicular  to  LH,  and  consequently  to  the  universal  me¬ 
ridian,  LM  and  FR,  each  parallel  to  AB,  and  LN  and  FG,  each 
parallel  to  TU.  Then,  as  in  a  preceding  article  (52),  DF  is  the 
distance  of  the  place  from  the  universal  meridian,  and  FG  is  its 
distance  from  the  plane  passing  through  the  centres  of  the  earth 
and  sun,  perpendicular  to  the  universal  meridian.  Also  PH  = 
PD  =  the  complement  of  the  latitude  of  the  place,  DLH  =  DPH 
=  the  hour  angle  from  noon,  and  TP  =  AC  =  the  sun’s  decli¬ 
nation.  Put, 

D  =  TP  as  the  sun’s  declination, 

U  =  DLH  =  the  hour  angle  from  noon, 

II  =  CH  =  the  reduced  latitude  of  the  place. 

Then  IIL  =  EH  sin  HEP  =  R  cos  H, 

and  DF  ==  DL  sin  DLH  =  PIL  sin  U  =  R  cos  H  sin  U. 

Hence  (54),  X  =  (P — p)  =  cos  H  sin  U.  (P — p)  A. 

R 


EL  =  EH  cos  HEP  =  R  sin  H, 

EM  =  EL  cos  TEP  =  R  sin  H  cos  D 
sin  (II  4-  D)  +  sin(H  —  D) 


(App.  16) 


ML  =  EL  sin  TEP  =  R  sin  II  sin  D, 

LF  =  DL  cos  DLH  =  IIL  cos  U  =  R  cos  II  cos  U. 
But  from  the  similar  triangles  FNL  and  EML,  we  have, 
EL  :  LM  :  :  FL  :  LN, 


CHAPTER  IX. 


135 


or  R  sin  H  :  R  sin  H  sin  D  :  :  R  cos  H  cos  U  :  MR 
1  :  sin  D  :  :  R  cos  H  cos  U  :  MR. 

Hence,  MR  =  R  cos  U  sin  D  cos  H 

=  R cos U. sm  (H  +  P)~slliH-~.D) (App.  17). 

£ 

When  the  latitude  of  the  place  and  declination  of  the  sun  ate 
both  north,  as  represented  in  the  figure,  ER  =  EM  —  MR.  It 
will  be  the  same,  when  the  latitude  and  declination  are  both 
south.  But  when  one  is  north  and  the  other  south,  ER  =  EM  + 
MR.  Therefore, 

FG  =  ER  =  EM  t  MR  =  R. sin  (H  +  D)  +  si»  (H  —  D 

A 

T  R  cos  U.  sin  (H  +  D)  —  Sin  (H  —  D) 

2 

Consequently  (54), 

y  _  sin  (H  +  D)  +  sin  (H  —  D)  (P__^ 

2 

^  sin  (H  +  D)  -  sin  (H  -  D)  cos  v  (p_  g 

2 

PROJECTION  OF  THE  SUN?S  CENTRE  ON  THE  CIRCLE  OF 
PROJECTION. 

56.  Let  AB,  Fig.  31,  be  the  line  in  which  the  plane,  through 
the  centres  of  the  sun  and  earth,  and  perpendicular  to  the  univer¬ 
sal  meridian,  intersects  the  circle  of  projection,  and  CD  the  inter¬ 
section  of  the  universal  meridian  with  the  same  circle.  With  the 
centre  C,  and  a  radius  equal  to  P  —  p,  describe  the  semicircle 
ADB,  to  represent  the  northern  half  of  the  circle  of  projection. 
With  a  sector,*  make  AE  and  BF,  each  equal  to  H,  the  reduced 

*  The  Sector  is  an  instrument,  generally  made  of  ivory  or  box  wood, 
about  a  foot  in  length,  with  a  joint  in  the  middle.  There  are  several  lines  on 
each  side  of  it.  But  the  only  one,  we  shall  notice,  is  the  line  of  chords,  which 
is  used  to  lay  off  a  given  number  of  degrees  on  the  arc  of  a  circle.  This  line 
is  marked  with  the  letter  C.  It  consists  of  two  lines,  running  each  way  from 
the  centre  of  the  joint  to  near  the  ends  of  the  instrument;  each  line  being 
divided  into  60  parts  or  degrees,  and  each  degree  subdivided  into  halves, 
At  the  60  on  each  line,  there  is  a  brass  pin  with  a  small  puncture. 


136 


ASTRONOMY, 


latitude  of  the  place,  and  make  EG,  El,  FH,  and  FK,  each 
equal  to  D,  the  sun’s  declination.  Join  GH,  EF,  and  IK,  and 
bisect  Rv  in  N.  Through  N,  draw  LNM  parallel  to  EF.  Make 
NO  equal  to  wE,  and  with  the  centre  N,  and  radii  NO  and  NR, 
describe  the  semicircles  PYO  and  tRQ.  With  a  sector,  make 
YW  equal  to  U,  the  hour  angle  from  noon.  Join  WN,  draw  WV 
parallel  to  CD,  and  through  T,  draw  STU  parallel  to  PO,  meet¬ 
ing  WY  in  U. 

In  this  construction,  the  sun’s  declination  is  supposed  to  be 
north,  and  the  time  in  the  afternoon.  When  the  declination  is 
south,  the  semicircle  ZRQ  must  be  on  the  upper  side  of  PO,  and 
WN  must  be  produced  to  meet  it  in  T.  When  the  time  is  before 
noon,  the  arc  YW  must  be  laid  off  from  \r  to  the  right  hand. 

Now,  NO  =  10E  =  w¥  =  CF  sin  FCw  =  cos  II.  (P  —  p)7 
SU  =  NV  =  NW  cos  WNL  =  NO  sin  U 
=  cos  H  sin  U.(P  —  p). 

Hence,  (55.  A),  SU  =  X. 

Also,  Cu  =  CH  cos  v  CH  =  sin  IICB.  (P — p) 

=  sin  (H  -f  D).  (P  —  p),- 
CR  =  CK  cos  RCK  =  sin  KCB.  (P  — p) 

=  sin  (H — D).  (P-rt, 

fixr  Cv  -f  CR  sin  (H  D)  -|-  sin  (H  D)  /p  v 

2  2  * v  Fh 

RN  ~~  s*n  _ p ^ 

2  2 


To  lay  off  any  number  of  degrees  on  the  arc  of  a  given  circle,  take  the 
radius  of  the  circle  in  the  dividers,  and  setting  one  foot  in  the  puncture  at 
the  end  of  one  line,  open  the  sector  till  the  other  foot  of  the  dividers  just 
reaches  to  the  puncture  at  the  end  of  the  other  line.  Then  setting  one  foot 
of  the  dividers  to  the  given  number  of  degrees  on  one  line,  open  them  till 
the  other  foot  reaches  to  the  same  number  of  degrees  on  the  other  line. 
This  distance,  applied  as  a  chord  to  the  arc,  will  intercept  the  given  number 
of  degrees. 

"  To  measure  a  given  arc,  open  the  sector  as  before.  Then,  taking  the 
chord  of  the  arc  in  the  dividers,  apply  them  to  the  line  of  chords,  moving 
them  without  changing  their  opening,  till  each  foot  is  at  the  same  number  of 
degrees  on  each  line.  This  number  of  degrees  will  be  the  measure  of  the 
given  arc. 


CHAPTER  XI* 


437 


•  NS  ==  NT  cos  YNW  =  NR  cos  U 
=  sjn(H  +  D)-sin(H-D),  cqs n (p _p)> 

CS  ==  CN  t  NS=  sin(H  i-  D)  i-  sin  (H  D) 

2 

t  —  iS.:~  P)~g'°  (H~ D)  cos U. (P— p). 

Hence  (55.  B),  CS  =  Y. 

Since  US,  the  distance  of  the  point  U,  from  the  universal  me* 
ridian,  is  equal  to  X,  and  *CS,  its  distance  from  the  plane,  per* 
pendicular  to  the  universal  meridian,  and  passing  through  the 
centres  of  the  earth  and  sun,  is  equal  to  Y,  the  point  U  is  the 
projection  of  the  sun’s  centre,  for  the  spectator  at  the  given  time, 
and  place  (54). 

57.  It  is  evident  from  the  construction,  that  the  semicircles 
PYO  and  fRQ,  depend  only  on  P  —  p,  the  difference  of  the  ho¬ 
rizontal  parallaxes  of  the  moon  and  sun,  D  the  sun’s  declination, 
and  H  the  latitude  of  the  place.  Therefore.,  since  P  —  p  and  D 
may  be  considered  as  constant  during  the  continuance  of  the 
eclipse,  the  same  semicircles  will  answer  for  finding  the  point  U 
at  any  other  time,  within  that  interval. 

58.  When  it  is  required  to  find  the  projection  of  the  sun’s 
centre  for  several  different  times,  on  the  same  figure,  it  is  better 
to  omit  drawing  the  lines  WN  and  UTS,  and  instead,  to  lay  the 
edge  of  a  ruler  from  W  to  N,  make  a  mark  at  T  on  the  semicircle 
fRQ,  and  draw  Ts  parallel  to  CD.  The  distance  Ts,  applied  on 
YW,  from  V  to  U,  will  give  the  point  U  the  same  as  before. 

59.  From  the  similar  triangles  WNV  and  TNS,  we  hafe 
WN  :  TN  :  :  WV  :  NS.  But  WN  =  NO,  TN  =  NR,  and 
NS  =  UV.  Therefore,  NO  :  NR  :  :  WV  :  UV.  Hence  (Conic 
Sections  ,  the  point  U  is  in  an  ellipse,  of  which  PO  is  the  trans¬ 
verse  axis,  and  Rv  the  conjugate.  On  account  of  the  earth’s 
diurnal  motion,  the  position  of  U  is  continually  changing.  But  it 
is  evident,  that  as  a  long  as  P  —  p  and  D  may  be  considered 
constant,  its  Path  will  be  the  ellipse  described  about  the  axes  PQ 
and  Rv. 


49 


138 


ASTRONOMY. 


POSITION  OF  THE  MOON?S  RELATIVE  ORBIT  ON  THE 
CIRCLE  OF  PROJECTION. 

60.  Make  Da  and  Db  each  equal  23°  28',  the  obliquity  of  the 
ecliptic;  join  ab  and  on  it  describe  the  semicircle  adb.  From  b 
lay  off  the  arc  bet  equal  to  the  sun’s  longitude.  When  the  longi¬ 
tude  exceeds  6  signs,  its  excess  above  6  signs  must  be  laid  off  from 
a  to  d.  Draw  Dm  perpendicular  to  a&,  and  through  m,  draw 
CmZ. 

Since  the  universal  meridian  passes  through  the  poles  of  the 
earth  (50),  it  coincides  with  a  circle  of  declination.  Hence  CD, 
its  intersection  with  the  circle  of  projection,  may  be  considered  as 
the  arc  of  a  declination  circle.  In  like  manner,  the  line  in  which  a 
circle  of  latitude  passing  through  the  centre  of  the  sun,  intersects 
the  circle  of  projection,  may  be  considered  as  the  arc  of  a  circle 
of  latitude.  The  angle  contained  between  these  lines  will  be  the 
angle  of  position  for  the  sun  (6.21).  Put, 

L  =  bd  =  sun’s  longitude, 
a  =  Db  =  obliquity  of  the  ecliptic, 

S  =  the  angle  of  position. 

Then  bn  =  C b  sin  b  CD  =  Cb  sin 

mn  =  nd  cos  bnd  =  bn  cos  L  =  C b  sin  »  cos  L. 

Cn  =  Cb  cos  b  CD  =  Cb  cos  <y, 

tan  DCZ  =  ~  =  9  sin  *  C0I±  =  tan  •  cos  L 
Cn  Cb  cos  a 

But  (6.21.C),  tan  S  =  tan  u  cos  L. 

Hence,  tan  DCZ  ==  tan  S,  or  DCZ  =  S. 

•Therefore  CZ  makes  with  CD,  the  angle  DCZ  equal  to  S5 
the  angle  of  position.  It  is  also  evident,  that  by  the  construction, 
CZ  will  fall  to  the  right  hand,  or  west  of  CD,  when  the  longitude 
is  less  than  90°  or  more  than  270°,  and  to  the  left  hand  or  east, 
when  the  longitude  is  more  than  90°  and  less  than  270°.  Hence 
CZ  is  a  circle  of  latitude,  the  plane  of  which  passes  through  the 
sun’s  centre. 

61.  Having  CZ  the  circle  of  latitude,  the  moon’s  relative  orbit 
may  be  drawn,  and  the  places  of  the  moon’s  centre  at  different 


CHAPTER  XI. 


139 


hours,  be  determined  in  the  same  manner  as  in  an  eclipse  of  the 
moon  (37  and  43),  only  using  the  time  of  conjunction  instead  of  the 
time  of  opposition.  Let  pq  be  the  moon’s  relative  orbit;  and  let 
u  be  the  place  of  the  moon’s  centre  at  the  same  time  the  projection 
of  the  sun’s  centre  is  at  U.  If  the  distance  Utt  is  less  than  the 
apparent  semi-diameters  of  the  sun  and  moon,  a  part  of  the  sun,  at. 
least,  is  then  eclipsed. 

62.  Considering  the  earth  a  sphere,  a  vertical  line  (4.4)  and 
consequently  a  verticlc  circle, ^at  a  given  place,  will  pass  through 
the  centre.  Hence,  since  C  is  in  the  straight  line  joining  the 
centres  of  the  sun  and  earth,  CU  is  the  intersection  of  the  circle 
of  projection  with  a  verticle  circle  passing  through  the  centre  of 
the  sun.  The  position  of  the  moon’s  centre,  with  regard  to  this 
circle,  is  therefore  determined, 

CONSTRUCTION  OF  AN  ECLIPSE  OF  THE  SUN. 

r  63.  The  construction,  Fig.  32,  is  for  an  eclipse  of  the  sun  on 
the  27th  of  August,  1821,  in  the  morning,  and  it  is  adapted  to  the 
meridian  and  latitude  of  Philadelphia.  The  points  7,  8,  9,  10, 
and  1 1  on  the  line  pq ,  represent  the  situation  of  the  moon’s  cen¬ 
tre  on  the  plane  of  projection,  at  those  hours  respectively.  The 
other  points  7,  8,  9,  10  and  11,  represent  the  projections  of  the 
sun’s  centre  at  the  same  times.  The  lines  7c,  8e,  9g,  lOfe,  and 
life  are  drawn  parallel  to  AB,  meeting  the  lines  Vile,  VIIIs, 
IXv,  Xw,  and  XI®,  in  the  points  c,  c,  g ,  h,  and  k. 

Draw  a  right  line  MN,  Fig.  33,  and  in  it  take  any  point  S,  to 
represent  a  fixed  position  of  the  sun’s  centre.  Then  taking  7c, 
the  distance  from  the  moon’s  centre  at  7  o’clock  to  the  line  Vile, 
lay  it  from  S  to  the  right  hand  to  c.  In  like  manner,  make  the 
distances  Se,  S g,  S/i,  and  Sfc,  in  the  line  MN,  respectively  equal  to 
the  distances  8e,  9g,  \0h.  and  life,  observing  that  each  distance  is 
to  be  laid  off  to  the  right  or  left  of  the  point  S,  according  as  the  moon’s 
centre  is  to  the  right  or  left  of  the  projection  of  the  sun’s  centre. 
Draw  the  lines  c7,  e8,  g9,  hi 0  and  fell,  perpendicular  to  MN. 
Make  c7  equal  to  the  distance  c7  on  the  line  passing  through  the 
projection  of  the  syn’s  centre  at  7  o’clock.  Also  make  c8,  g*9, 
MO,  and  fell,  respectively  equal  to  the  corresponding  distances 


140 


ASTRONOMY. 


in  Fig.  32,  observing  that  each  must  be  placed  above  or  below 
the  line  MN,  according  as  the  moon’s  centre  is  higher  or  lower 
than  the  projection  of  the  sun’s  centre.  Then  the  points  7,  8,  9, 
10,  and  11,  Fig.  33,  will  represent  the  positions  of  the  moon’s 
centre,  at  those  hours,  with  regard  to  S,  the  sun’s  centre.  Join 
*he  points  7,8,  8,9,  9,10,  and  10,11,  and  the  broken  line  thus 
formed,  which  will  deviate  but  little  from  one  right  line,  will  repre¬ 
sent  very  nearly  the  apparent  relative  orbit  of  the  moon.  With  the 
centre  S,  and  a  radius  equal  to  the  sum  of  the  apparent  semi¬ 
diameters  of  the  sun  and  moon,  describe  arcs  cutting  the  moon’s 
path  in  h  and  r,  which  will  be  the  positions  of  the  moon’s  centre 
at  the  beginning  and  end  of  the  eclipse.  From  S,  draw  Sq  per¬ 
pendicular  to  the  part  8,9,  of  the  moon’s  path;  then  q  will  be  the 
position  of  the  moon’s  centre,  when  it  is  nearest  to  the  centre  of 
the  sun,  and  consequently  when  the  eclipse  is  greatest.  If  the 
hour  spaces  7,8,  8,9,  and  10,11,  be  each  divided  into  quarters 
and  these  subdivided  into  three  equal  parts,  or  spaces  of  5  mi¬ 
nutes,  the  times  of  beginning,  greatest  obscuration,  and  end,  can 
be  easily  estimated. 

64,  The  hour  spaces  on  the  moon’s  apparent  path  are  not  equal, 
and  therefore  the  times  obtained  from  equal  divisions  of  them,  are 
not  quite  accurate.  The  error  from  this  cause  will  not,  how¬ 
ever,  exceed  one  or  two  minutes;  and  if  the  construction  be  made 
for  each  half  hour,  it  will  be  much  less. 

With  the  centre  S,  and  a  radius  equal  to  the  sun’s  apparent 
semi-diameter  describe  a  circle  to  represent  the  sun’s  disc;  and 
with  the  centre  #7,  and  a  radius  equal  to  the  moon’s  apparent  semi¬ 
diameter  describe  another  circle  to  represent  the  moon’s  disc. 
The  part  of  the  sun’s  disc,  that  is  intercepted  by  the  moon’s, 
shows  the  part  eclipsed.  If  S q  be  produced  to  in  and  w,  and  mn 
be  measured  by  the  scale,  used  in  the  construction,  the  quantity  of 
the  eclipse  will  evidently  be  obtained  by  this  proportion.  As  the 
sun's  apparent  diameter  :  mn  :  :  12  digits  :  the  digits  eclipsed. 

Let  w,  Fig.  32,  be  the  projection  of  the  sun’s  centre  at  the  time 
the  eclipse  commences.  Then  Cu  will  be  a  verticle  circle  passing 
through  the  sun’s  centre  at  that  time  (62).  Draw  SF,  Fig.  33, 
making  the  angle  FSM  equal  to  the  angle  ttCB;  then  v  is  the 


CHAPTER  XI. 


141 


sun’s  vertex.  Now  as  the  eclipse  commences  at  the  point  a  of 
the  sun’s  disc,  the  angle  vSa  expresses  the  angular  distance  from 
the  sun’s  vertex,  of  the  point  at  which  the  eclipse  commences. 
The  knowledge  of  this  angle  is  important  to  the  astronomer,  who 
wishes  to  observe  with  accuracy  the  commencement  of  an  eclipse 
of  the  sun.  Without  it  he  would  not  know  at  what  part  of  the 
edge  to  fix  his  attention,  while  waiting  to  see  the  first  impression. 

CALCULATION  OF  AN  ECLIPSE  OF  THE  SUN. 

67.  Let  B  designate  the  approximate  time  of  the  beginning  of 
the  eclipse,  found  by  construction  (63),  and  t  some  short  interval 
of  4  or  5  minutes.  Calculate  for  the  time  B  —  f,  by  means  of  as¬ 
tronomical  tables'  the  sun’s  longitude,  hourly  motion,  and  semi¬ 
diameter;  also  the  moon’s  longitude,  latitude,  horizontal  parallax, 
semidiameter,  and  hourly  motions  in  longitude  and  latitude.  Then 
as  our  object  is  to  obtain  the  difference  of  the  apparent  longitudes 
of  the  sun  and  moon,  and  the  moon’s  apparent  latitude,  in  order  to 
obtain  their  apparent  distance,  subtract  the  sun’s  horizontal  paral¬ 
lax,  from  the  reduced  horizontal  parallax  of  the  moon,  and  con¬ 
sidering  the  remainder  as  the  moon’s  parallax,  calculate  the  paral¬ 
lax  in  longitude  and  latitude  (10.  54  and  56),  using  the  reduced 
latitude  of  the  place*.  To  the  moon’s  true  longitude  and  latitude, 
apply,  respectively,  the  parallaxes  in  longitude  and  latitude  ac¬ 
cording  to  their  signs,  and  the  apparent  longitude  and  latitude  are 
obtained.  Take  the  difference  between  the  true  longitude  of  the 
sun  and  the  apparent  longitude  of  the  moon,  which  will  be  the 
moon’s  apparent  distance  from  the  sun,  in  longitude. 

68.  With  the  sun  and  moon’s  longitudes,  the  moon’s  latitude, 
and  their  hourly  motions  at  the  time  B  —  t,  find  the  longitudes  and 
the  moon’s  latitude  at  the  time  B  -j-  t.  For  this  latter  time,  cal¬ 
culate  the  parallaxes  in  longitude  and  latitude,  and  thence  deduce 


*  This  is  equivalent  to  considering  the  apparent  place  of  the  sun,  the  same 
as  the  true,  and  referring  the  whole  effect  of  parallax  to  the  moon.  It  is  not 
rigidly  exact.  For  we  virtually  calculate  the  sun’s  parallax  in  longitude  and 
latitude  by  making  use  of  the  moon’s  longitude  and  latitude,  instead  of  the 
sun’s.  But  on  account  of  the  small  quantity  of  the  sun’s  parallax,  and  the 
little  difference  in  the  longitudes  at  the  time  of  an  eclipse,  the  error  is  quite 
insensible. 


142 


ASTRONOMY 


the  apparent  distance  of  the  moon  from  the  sun  in  longitude,  and 
the  moons’s  apparent  latitude. 

69.  Let  EC,  Fig.  34,  be  a  part  of  the  ecliptic  and  S  the  place 
of  the  sun’s  centre,  which  we  shall  consider  fixed.  Let  SA  be 
the  apparent  distance  of  the  moon  from  the  sun  in  longitude,  and 
AD,  perpendicular  to  EC,  the  apparent  latitude,  at  the  time 
B  —  l.  Then  D  will  be  the  moon’s  apparent  place.  In  like 
manner  let  L  be  the  moon’s  apparent  place  at  the  time  B  -l  t. 
The  line  DL,  which  does  not  sensibly  differ  from  a  straight  line, 
represents  the  part  of  the  moon’s  apparent  relative  orbit,  passed 
through  during  2 f,  the  interval  between  the  times  B  —  t  and 
B  +  t. 

70.  Let  the  point  G  in  the  line  DL,  be  the  place  of  the  moon’s 
centre  at  the  true  time  of  beginning.  Put, 

it  s  SA  =  appar.  dist.  of  moon  from  sun  in  long,  at  the  time 
B  —  £, 

c  —  AD  =  moon’s  app.  latitude  at  the  time  B  —  t , 
m  =  AH  =  diff.  of  moon’s  app.  distances  from  the  sun  in  long, 
at  B  —  t  and  B  4-  £, 

n  =  wL  —  diff.  of  moon’s  apparent  latitudes  at  B  —  t  and  B  4-  t. 
s  =  SG  =  sum  of  app.  semi-diameter  of  sun  and  moon, 

I  =  angle  wDL  =  inclination  of  moon’s  app.  rel.  orbit, 
x  =  AF. 

Then  tan  I  =  tan  wDL  =  ~  =  -, 

\jw  m 

Gu  =  Dtf  tan  vDG  =  x  tan  I, 

SF  =  a  —  x ,  and  FG  =  AD  -f  Gi>  ==  c  +  x  tan  I, 
SF3  +  FG2  ==  SG2,  or  (a  —  x)2  +  (c  +  x  tan  I)  2=  s2 
a2  —  2ax  +  x2  +  c2  -f  2cx  tan  I  -f  x2  tan2  I  =  s2 

(B) 

(1  4.  tan2  I)  .a2  — 2  (a  —  c  tan  I)  x  =s2  —  a 2  — c2, 

2  2.  (a  —  ctan  I)  s2  —  a2 — c2 

*  1  +  tan2 1  1  -f  tan2 1  5 

2 _ 2.  (ft  —  c  tan  I)  ,  (a  —  ctanl)2  =  s2  —  a 2  —  c2 

1  -f  tan2 1  (1  +  tan2 1) 2  1  4- tan2 1 

(a  —  ctan I)  2 
(1  4.  tan2 1)  z 


CHAPTER  XI. 


143 


__  (s2  —  a*  —  c2).  (1  +  tan2 1)  +  (a  —  ctan  I)g 

( 1  -f  tan2 1)  2 

$2  4-  s2  tan2 1  —  (c2  4-  2  a  c  tan  I  4-  a2  tan2 1) 

;  (1  4-  tan2 1) 2 

s2.  (1  4-  tan2 1)  —  (c  4-  a  tan  I) 2 
=  '(1  +  tan2 1)  2  “ 

s2  _ (c  4-  a  tan  I)  2\ 

~  1  4-  tan2  I.'  V  s2.  (1  4-  tan2I )/ 

= s2.  cos2 1.  (i  -  (c+aty-c-°ii-1) 

„,_g-ctanl  —  4- cos  I,/  (,  _(i±ftanI)W_r\ 
1  +  tan2 1  v  s2  / 

/  ,  t\  o  t  ft.i  ,(\  (c  4-  a  tan  I)2  cos2 1 \ 

x  =  (a  —  c  tan  I)  cos2 1  ±  s.  cos  I  v/  ^1  —  ^ - —J. - J 

i  .  .  (c  4-  a  tan  I)  .cos  I 

Make  sin  0  =^T - L - 

s 

Then,  sin2<  =  £+  «tanI)2.cos2I 
’  s2 

a  j  ,  /i  (c  +  «tanD2cos2I\  /t  .  2  \  . 

And  -v/  —  Z - - j  =  ^1  —  sm20j=cos*. 

Hence,  x  =  (a  —  c  tan  I)  cos2 1  ±  s.  cos  I  cos  0. 

If  the  line  DL  were  produced,  it  is  evident  there  would  be 
another  point  in  it,  such  that  its  distance  from  S,  would  be  the 
same  with  that  of  the  point  G.  The  two  values  of  x  in  the  for¬ 
mula,  correspond  to  these  two  points.  Hence  the  value  of  a;  cor¬ 
responding  to  G,  is, 

x  =  (a  —  c  tan  I)  cos3  I  —  s.  cos  I  cos  0. 

When  the  moon’s  apparent  latitude  at  the  time  B  4-  t  is  less 

than  at  the  time  B  —  f,  we  must  make, 

.  .  (c  —  a  tan  I)  .cos  I  .  ,  „  , 

sin  0  =  A - 1 - ,  and  we  shall  have 

s 

x  =  (a  4-  c  tan  I)  .cos*  I  —  s .  cos  I  cos  0. 

2tx 

Now,  as  m  :  a  :  :2t : - =  interval  from  B  —  t  to  the  true 

m 

time  of  the  beginning.  Hence  if  B'  =  the  true  time  of  beginning, 
we  have, 


144 


ASTRONOMY. 


tan 


B  =  B  —  t 


l—l  sin  6  —  (c±a  tan  I)  cos  1 
m  s 

4-  ((flTc tan  I)  .cos8  I — s  cos  I  cos  6 


The  upper  signs  must  be  used  when  the  apparent  latitude  is 
increasing ,  and  themder,  when  it  is  decreasing.  The  same  is  to 
be  observed  in  the  three  following  articles. 

71.  The  end  of  the  eclipse  may  be  found  in  a  similar  manner. 
Let  E  be  the  approximate  time  of  the  end,  and  make  the  same 
calculations  for  E  —  t  and  E  4.  f,  as  for  B  —  t  and  B  4  t.  Then, 
if  a,  c,  m,n,s,  I  and  0,  designate  the  same  quantities  as  in  the  last 
article,  but  having  the  values  appertaining  to  the  end  of  the 
eclipse,  and  E'  =  the  true  time  of  end,  we  have, 

tan  1=2,  sine=^otanl)  cosl, 
m  s 

E'  =  E  — t  -f  (5.  cos  I  cos  0  —  (a±  c  tan  I)  cos2  i)  *  ^  \ 

72.  To  find  the  true  time  of  greatest  obscuration,  let  G  be  the 
approximate  time,  and  calculate  as  before,  for  the  times  G  —  t 
and  G  -ft,  the  apparent  distances  of  the  moon  from  the  sun,  in 
longitude,  and  the  moon’s  apparent  latitudes.  Let  SK  be  the  ap¬ 
parent  distance  of  the  moon  from  the  sun  in  longitude  at  the  time 
G  —  t,  SN,  at  the  time  G  +  £,  and  KI,  NP  perpendiculars  to  EC, 
the  corresponding  latitudes.  Then  if  S q  be  drawn  perpendicular 
to  IP,  q  will  be  the  place  of  the  moon,  when  the  apparent  distance 
of  the  centres  is  least.  Let  x—  KV,  and  a,  c,  m/n,  s  and  I,  de¬ 
signate  the  same  quantit  ies  as  before,  but  having  the  values  apper¬ 
taining  to  the  middle. 

From  the  similar  triangles  PIQ  and  SgV,  we  have, 

IQ  :  PQ  :  :  Vq  :  SV. 

But  IQ  =  m,  PQ  =  w,  \q  =  c  4  x  tan  I,  and  SV  —  a  —  x. 
Hence,  m  :  n  :  :  c  +  x  tan  I  :  :  a  —  x , 

?—*  =  2  =  tan  I, 

c  -f  x  tan  I  m 

x—  ~ C-tan  *  =  (a —  c  tan  I)  .cos*  I 
1  4-  tan2  I  v  ' 


CHAPTER  XI. 


145 


When  the  apparent  latitude  is  increasing, 

x  =  (a  4-  c  tan  I)  cos2  I. 

Hence,  if  G'  =  the  true  time  of  greatest  obscuration,  we  have, 

tanl=^,  G=G —  t  4-  (a=F  c  tan  I)  .cos2  I  — . 
m  m 

73.  To  find  Sq,  the  nearest  distance  of  the  centres,  we  have 

(72), 

Yq  =  c  ±  x  tan  I  =  c  ±  tan  I.  (a^fc  tan  I)  cos2  I 
—  c  ±  sin  I  cos  I.  (a  c  tan  I). 


Hence,  S q 


Yq 


cos  I  cos  I 


±  sin  I.  (a  qF  c  tan  I) 


Now  to  find  the  quantity  of  the  eclipse,  if  <^=  the  sun’s  appa¬ 
rent  semidiameter,  and  d  —  the  moon’s  apparent  semidiameter  at 

the  time  of  the  greatest  obscuration,  we  have,  Fig.  33. 

1  '  > 

mn  —  Sq  -f  qn  4-  S m  =  S 7  +  ^  —  Sq  +  d—Sq  =  f  +  d  —  Sq 


=  s — Sq  = 
Digits  Eclipsed 


cos  I 


=F  sin  I.(atc tan  I).  Hence, 


*  ( s  — - Tasini. (aq=c tan I) \ 

.mn  _  6.  \  cos  I  '  J ) 


74.  When  Sq,  the  nearest  apparent  distance  of  the  centres,  is 
less  than  the  difference,  between  $  and  d,  the  eclipse  will  be  either 
annular  or  total.  It  will  be  annular  if  l  be  greater  than  d,  and 
total  if  d  be  greater  than  £  The  time  when  an  eclipse  commences 
or  ceases  to  be  total  or  annular  may  be  found  by  the  same  for¬ 
mulae  as  the  beginning  or  end  of  the  eclipse,  only  making  s  =  to 
the  difference  of  and  d,  and  giving  to  a,  c,  m,  &c.  the  values 
which  they  have  in  finding  the  time  of  greatest  obscuration. 

75.  The  greatest  duration  of  an  annular  eclipse  at  any  place  is 
about  12  minutes,  and  of  a  total  eclipse  about  8  minutes.  In  a  total 
eclipse  the  obscurity  is  such  as  to  render  the  principal  stars  and 
most  conspicuous  planets  distinctly  visible. 

76.  Let  S b,  Fig.  34,  be  a  circle  of  latitude,  Sd  a  declination 
circle,  and  Sh  a  vertical  circle,  all  passing  through  the  sun’s  centre. 
Then  bSd  will  be  the  angle  of  position,  and  dSh  the  angle  con- 

20 


146 


ASTRONOMY. 


tained  by  the  declination  circle  and  vertical  circle,  passing  through 
the  sun.  Put  L  =  the  sun’s  longitude  at  the  beginning  of  the 
eclipse,  A  =  the  sun’s  distance  from  the  north  pole  of  the  equator, 
a  —  SF  =  the  apparent  difference  of  the  sun  and  moon’s  longi¬ 
tudes,  c  =  the  moon’s  apparent  latitude,  II  =  the  latitude  of  the 
place,  a  —  the  obliquity  of  the  ecliptic,  and  U  =  the  hour  angle 
from  noon.  Then  (6.  20  and  21) 

cos  A  =  sin  L  sin  */,  and  tan  bSd  —  cos  L  tan  a  (C) 

In  the  triangle  PS Z,  Fig .  18,  we  have  (App.  87), 

crq  £  _  co*  s*n  PS  —  cos  ZPS  cos  PS 
*  snTzps 

tan  H  sin  A  —  cos  U  cos  A 
sin  U 

Hence,  Fig.  34,  tan  dSk  =  sin  a  -  cos  Ucos^ 

sinU 

Also  tan  FS6  =  55  =  £  -  .  (E) 

IS  d 

The  angle  bSd  will  be  to  the  left  of  S 6,  when  the  sun’s  longi¬ 
tude  is  less  than  90°,  or  more  than  270°,  and  to  the  right  when  it  is- 
between  90°  and  270°.  The  angle  d  S/i  will  be  to  the  right  of  S d7 
in  the  forenoon,  and  to  the  left  in  the  afternoon.  By  attending  to 
these  circumstances,  and  adding  or  subtracting  accordingly,  the  an¬ 
gle  b$h,  and  consequently  ES h,  becomes  known.  Thence,  by  ap¬ 
plying  the  angle  FSG,  we  have  the  angle  /iSG,  contained  between 
the  vertical  circle  passing  through  the  sun’s  centre,  and  the  line 
joining  the  centres  of  the  sun  and  moon,  at  the  beginning  of  the 
eclipse. 

77.  Supposing  the  latitude  of  the  place  for  which  the  calcula¬ 
tion  is  made,  to  be  truly  known,  and  also  its  longitude  from  the 
place  for  which  the  solar  and  lunar  tables  are  computed,  the  results 
obtained  from  the  preceding  formulas,  when  the  calculations  are 
carefully  performed,  will  have  an  accuracy  corresponding  with  that 
of  the  tables  themselves,  or  very  nearly  so.  But  the  best  of  these 
tables  are  liable  to  errors  of  a  few  seconds.  Consequently  the 
times  obtained,  will  be  liable  to  small  errors,  depending  on  the 
former. 


CHAPTER  XT. 


147 


78.  Except  in  cases,  when  the  greatest  precision  is  required  in 
the  results,  it  will  not  be  necessary  to  calculate  the  longitudes  of 
sun  and  moon,  and  the  moon’s  latitude,  from  the  tables,  only  for 
the  time  G  —  t.  The  longitudes  and  latitude  at  the  times  B  —  l 
and  E  —  £,  may  be  found  from  the  former  by  means  of  their  hourly 
motions.  It  will  also  be  sufficient  to  calculate  the  moon’s  appa 
rent  longitudes  and  latitudes  for  the  times  B —  £,  G — t  and 
E  —  t.  Then  for  the  beginning  of  the  eclipse  we  may  take  m 
=  moon’s  apparent  relative  motion  in  longitude  during  the  time 
(G  —  t)  —  (B  —  f),  or  which  is  the  same  G  —  B,  and  n  = 
moon’s  apparent  motion  in  latitude  during  the  same  time.  For  the 
end  we  may  give  to  m  and  w,  the  values  of  the  same  quantities,  for 
the  time  E  —  G.  And  for  the  greatest  obscuration  we  may  give 
to  them  the  values  of  those  quantities  for  the  time  E  —  B.  With 
these  values  of  m  and  w,  we  obtain  very  nearly  the  inclinations  of 
the  apparent,  relative  orbit  at  the  times  of  beginning,  end,  and 
greatest  obscuration . 

As  the  value  of  x,  in  the  equation  B  (70),  must  be  small,  its 
square  may  be  omitted  with  but  little  error.  We  shall  then  have, 


a2  —  2 ax  4.  c2  4  2cx  tan  I  =  s2, 
2.  (ft  —  c  tan  I)  .x  —  a2  4  c2  —  s2 , 
 ct2  -f  c2  —  s2 


2.  (a  —  ctan  I)’ 

Hence  B  =B  — 1+  (-?—+. c*  ~~  ffj •..G~B2 

2m  (  a  ^  c  tan  I) 

E'  =  E  —  t  4  (ft2  4-  c2  »  (E  G) 

2m  (a  ±  c  tan  I) 

q, _ q _ t  [ft  T  c  tan  I)  cos2  I]  .  E — B) 


In  each  formulas  the  upper  sign  is  to  be  used  wheAthe  apparent 
latitude  is  increasing ,  and  the  lower  when  it  is  decreasing. 

79.  Instead  of  finding  the  approximate  times  of  beginning  and 
end  by  construction,  we  may,  though  with  more  labour,  perform 
the  whole  by  calculation.  Thus,  let  the  sun’s  longitude  and  the 
moon’s  apparent  longitude  and  latitude  be  calculated  for  the  time 
of  new  moon.  From  these  longitudes  we  know,  whether  the  Jip- 


148 


ASTRONOMY. 


parent  Ecliptic  Conjunction ,  that  is,  the  circumstance  of  the  appa¬ 
rent  longitude  of  the  moon  being  the  same  with  the  sun’s  longitude, 
has  place  before  or  after  new  moon  For  a  time  an  hour  or  two, 
earlier  or  later,  than  the  new  moon,  according  as  the  apparent  con¬ 
junction  is  before  or  after,  again  calculate  the  sun’s  longitude  and 
moon’s  apparent  longitude  and  latitude.  Then  considering  the 
apparent  relative  orbit  of  the  moon  as  a  straight  line  passing  through 
the  apparent  positions  of  the  moon  with  regard  to  the  sun,  at  those 
times,  we  can  obtain  the  approximate  times  of  beginning,  greatest 
obscuration  and  end,  nearly  in  the  same  manner  as  the  beginning, 
middle,  and  end  of  a  lunar  eclipse. 

PATH  OF  A  CENTRAL  ECLIPSE  OF  THE  SUN. 

80.  The  latitude  and  longitude  of  the  place,  to  which  the  sun 
is  centrally  eclipsed,  at  a  given  time  during  the  continuance  of 
the  central  eclipse,  may  be  easily  determined  with  considerable 
accuracy,  by  means  of  a  geometrical  construction. 

If  we  suppose  the  circle  ATBU,  Fig .  30.  to  be  described  with 
a  radius  equal  to  P  — p ,  it  is  evident  (54  and  55),  that  FD  =  X, 
and  ER  =  FG  =  Y.  Then  in  the  right  angled  triangle  EFD, 
we  have  EF  =  (ED2  —  FD2)  =  v/  [(P  — p)  2  — X2]. 
Now  if  X  and  Y  are  known  for  any  given  time,  EF  and  ER  are 
known,  and  consequently  the  position  of  the  point  F.  The  po¬ 
sition  of  PE  is  determined  by  the  sun’s  declination,  and  therefore 
that  of  the  line  FIFL,  which  passes  through  F,  perpendicular  to 
PE.  In  the  right  angled  triangle  ELH,  EL  and  EH  being 
known,  the  angle  PEH,  the  complement  of  the  latitude  of  the 
place,  is  also  known.  In  the  right  angled  triangle,  DLF,  FL  and 
FD  being  known,  the  angle  DLH,  which  is  the  hour  angle  from 
noon,  is  known. 

81.  Let  AUBT,  Fig.  35,  be  the  circle  of  projection,  described 
with  the  radius  P  —  p,  TU  the  intersection  of  the  universal  me  ¬ 
ridian  with  this  circle,  AEB  perpendicular  to  TU,  and  pq  the 
moon’s  relative  orbit  (60).  If  D  be  the  place  of  the  moon’s  cen¬ 
tre  at  a  given  time,  on  a  given  meridian,  it  is  manifest  the  place 
which  w  ould  have  the  projection  of  the  sun’s  centre  also  at  D, 
would  then  have  a  central  eclipse.  Make  TP  =  the  sun’s  de- 


CHAPTER  XI. 


149 


di nation,  laying  it  to  the  left  of  T,  when  the  declination  is  north , 
but  to  the  right ,  when  the  declination  is  south ,  and  draw  PEP'. 
Through  D,  draw  MDG  parallel  to  TU,  and  DF  parallel  to  AB. 
Then  KM  =  y/  (EM2  —  EK2 )  =  ^  [  P  _p)»  —**].  With 
the  centre  E,  and  a  radius  equal  to  KM,  describe  an  arc,  cutting 
DF,  in  F,  and  through  F,  draw  LFH,  perpendicular  to  PP.  Then 
PEH  is  the  complement  of  the  latitude  of  the  place  (80.). 

Make  KG  =  FL,  and  through  G,  draw  EGI.  Then  because 
NG  =  EK  =  X,  and  EN  =  G£  ==  FL,  UEI  is  the  hour  angle, 
from  the  universal  meridian  at  the  required  place  (SO).  The  place 
will  be  to  the  west  or  east ,  of  the  universal  meridian,  according 
as  the  point  D  is  to  the  right  or  left  of  TU.  Now  for  the  place, 
for  which  the  construction  is  made,  the  time  that  the  moon’s  cen¬ 
tre  is  at  D,  and  consequently  the  distance  of  the  place  from  the 
universal  meridian,  is  known.  Hence  for  the  same  instant,  wc 
know  the  distance,  in  time,  of  the  given  place,  and  required  place, 
from  the  universal  meridian;  and  by  taking  the  difference,  or 
sum  of  these  distances,  according  as  they  are  on  the  same,  or  dif¬ 
ferent  sides  of  the  universal  meridian,  the  longitude  in  time,  of 
the  required  place,  from  the  given  place,  becomes  known. 

By  making  the  construction  for  every  15  or  20  minutes,  during 
the  continuance  of  the  central  eclipse,  we  shall  have  the  latitudes 
and  longitudes  of  a  series  of  places,  at  which  the  eclipse  will  be 
central.  A  curve  line,  drawn  on  a  map  or  globe,  through  those 
places,  will  represent,  what  is  called  the  Path  of  the  Central 
Eclipse. 

82.  By  a  process  but  little  different  from  the  preceding,  the 
longitudes  and  latitudes  of  those  places  that  will  have  the  eclipse 
of  a  given  magnitude,  for  instance  6  or  9  digits,  may  be  obtained. 

The  places  at  which  an  eclipse  will  be  central  or  of  a  given 
magnitude,  may  be  determined  more  accurately  by  calculation. 
The  methods  of  making  these  calculations,  are  given  in  our  larger 
treatises  on  astronomy. 


OCCULTATIONS. 

83.  If  at  the  time  of  mean  conjunction  of  the  moon 
anti  a  star,  that  is,  when  the  moon’s  mean  longitude  is 


150 


ASTRONOMY. 


the  same  with  the  longitude  of  the  star,  their  difference 
of  latitude  exceed  1°  37'  there  can  not  be  an  occultation; 
but  if  the  difference  be  less  than  51',  there  must  be  an 
occultation  somewhere  on  the  earth.  Between  these 
limits  there  is  a  doubt,  which  can  only  be  removed  by 
the  calculation  of  the  moon’s  true  place. 

84.  The  construction  for  an  occultation,  is  nearly  the  same  as 
for  an  eclipse  of  the  sun.  There  is  however  a  difference  in  some 
parts.  The  radius  CB,  Fig.  31,  must  be  equal  P,  for  a  star, 
and  P — p  for  a  planet,  p  being  the  horizontal  parallax  of  the 
planet.  The  arcs  FH,  FK,  EG  and  El  must  be  equal  to  the  de¬ 
clination,  and  bd  to  the  longitude,  of  the  star  or  planet.  In  de¬ 
termining  the  projected  place  of  the  star,  for  a  given  time,  we 
must  use  the  hour  angle  corresponding  to  the  difference  between 
the  given  time,  and  the  time  the  star  is  on  the  meridian. 

To  get  the  position  of  the  circle  of  latitude,  lay  off  the  declina¬ 
tion  of  the  star  or  planet,  from  D  to  &',  and  draw  b'c  parallel  to 
ah.  With  the  centre  C  and  radius  Cc,  describe  the  arc  cm',  cut¬ 
ting  dm,  produced  if  necessary,  in  m'.  Join  Cm'  which  will  be 
the  circle  of  latitude,  as  is  easily  deduced  from  the  expression  for 
the  angle  of  position  (6.21.  B.) 

The  distance  O'  on  the  circle  of  latitude,  must  be  equal  to  the 
difference  or  sum,  of  the  latitude  of  the  moon,  and  that  of  the  star 
or  planet,  according  as  they  are  of  the  same,  or  of  different  names. 
It  must  be  placed  above  C,  when  the  moon  is  to  the  north  of  the 
other  body,  but  below,  when  it  is  to  the  south. 

For  a  star,  the  moon’s  motion  in  longitude,  will  be  its  rela- 
live  motion  in  longitude.  For  a  planet  the  moon’s  relative  mo¬ 
tion  in  longitude  is  obtained  by  subtracting  the  motion  of  the  planet 
when  direct,  and  adding  it  when  retrograde.* 

In  Fig.  33,  it  is  evident,  that  for  a  star  we  must  take  S b  and 
Sr,  each  equal  to  the  apparent  semidiameter  of  the  moon;  and  for 
a  planet,  equal  to  the  sum  of  the  apparent  semidiameters  of  the 
moon  and  planet. 

*  The  apparent  motion  of  a  planet  is  sometimes  retrograde.  This  cir¬ 
cumstance  will  be  more  particularly  noticed  in  the  next  chapter. 


CHAPTER  XI. 


151 


85.  When  considerable  accuracy  is  required,  the  moon’s  rela¬ 
tive  motion  on  the  ecliptic  must  be  reduced  to  its  motion  on  a  pa¬ 
rallel  to  the  ecliptic,  passing  through  the  star  or  planet.  This 
is  done  by  multiplying  the  relative  motion  in  longitude  by  the 
cosine  of  the  latitude  of  the  star. or  planet.  For  if  AB,  Fig.  36, 
be  an  arc  of  the  ecliptic,  and  DE  the  corresponding  arc  of  a  circle, 
parallel  to  it,  we  have, 

BC  :  EF  :  :  AB  :  DE  = 

BC 

But,  EF  =  C a  —  BC  cos  BCE  =  BC  cos  BE. 

Hence,  DE  =  AB-  BCcos  BE  =  AB  cos  BE 
’  BC 

86.  The  difference  between  the  calculation  of  an  occultation 
and  that  of  an  eclipse  of  the  sun,  is  easily  deduced  from  what  has 
been  said  in  the  two  preceding  articles. 

87.  Observations  of  an  eclipse  of  the  sun  or  of  an 
occultation  of  a  star,  made  at  places,  whose  longitudes 
and  latitudes  are  correctly  known,  furnish  means  of  de¬ 
termining  the  errors  of  the  tables  at  the  time;  and  they 
are  frequently  used  for  that  purpose,  particularly  those 
of  an  occultation.  The  positions  of  many  of  the  stars 
are  determined  with  great  precision,  and  the  moon’s 
parallax  and  apparent  diameter  are  very  accurately 
known.  But  the  moon’s  longitude  and^  latitude,  com¬ 
puted  from  the  best  lunar  tables  are  liable  to  errors  of 
several  seconds.  Hence  if  the  observed  time  of  be¬ 
ginning  or  end  of  an  occultation,  of  a  star  whose  posi¬ 
tion  is  well  determined,  does  not  agree  with  the  time, 
obtained  by  calculation,  the  difference  must  depend  on 
errors  in  the  computed  longitude  and  latitude  of  the 
moon. 

88.  It  is  evident  from  the  formulae  for  computing  the  parallaxes 
in  longitude  and  latitude,  that  those  parallaxes  are  not  sensibly  ef- 


152 


ASTRONOMY. 


fected  by  small  errors  in  the  longitude  and  latitude.  The  errors 
in  the  apparent  longitude  and  latitude,  may  therefore  be  considered 
the  same  as  those  in  the  true. 

89.  Let  a  and  c  be  the  apparent  distances  of  the  moon  from  the 
star,  in  longitude  and  latitude,  respectively,  as  obtained  by  calcu¬ 
lation  for  the  observed  time  of  beginning,  and  l  be  the  latitude  of 
the  star.  Also  let  x  =  the  error  in  the  moon’s  longitude,  and  y 
=  the  error  in  the  latitude.  Then  a  +  x  and  c  y  will  be  the 
true,  apparent  distances  of  the  moon  from  the  star  in  longitude 
and  latitude,  at  the  observed  time  of  beginning.*  Consequently, 
if  s  =  the  moon’s  apparent  semidiameter,  we  have, 

(a  +  x  2.  cos2  Z  +  (c  +  y)  2  =  s2, 

or,  a2  cos2Z  +  2  ax  cos2 1  -f-  x 2.  cos2 1  -f  c2  2  cy  -\-y2  =  s2. 

Now  as  x  and  y  are  small  quantities,  the  terms  involving  their 
squares,  may  be  omitted.  Hence,  if  §  (s2  —  a2  cos2 1  —  c2)  =  e, 
we  have, 

a  cos2 1.  x  cy  —  e. 

As  the  errors  x  and  y  will  not  sensibly  change  during  the  con¬ 
tinuance  of  an  occultation,  another  similar  equation  may  be  ob¬ 
tained  from  an  observation  of  the  end,  and  a  calculation  for  that 
time.  Thus, 

a',  cos2 1.  x  +  c'y  —  e'. 

From  these  two  equations  the  values  of  x  and  y  are  easily 
found. 

90.  The  errors  of  the  lunar  tables,  in  longitude  and 
latitude,  may  also  be  determined,  by  observing  the 
moon’s  right  ascension  and  declination,  either  at  the 
place  for  which  the  tables  are  constructed,  or  at  any 
other  whose  longitude  and  latitude  are  accurately 
known.  From  the  observed  right  ascension  and  de- 

*  The  signs  of  x  and  y ,  are  both  put  affirmative.  If  either  or  both  of  them 
ought  to  be  negative,  it  will  be  determined  by  the  calculation. 


CHAPTER  XI. 


153 


clination,  the  moon’s  longitude  and  latitude  may  be 
calculated  (0.19),  and  thence  the  errors  ascertained. 

91.  In  calculating  the  moon’s  longitude  and  latitude 
for  any  instant  of  time,  as  reckoned  at  a  given  place 
on  a  meridian,  different  from  that  for  which  the  tables 
arc  constructed,  we  most  reduce  the  given  time,  to  the 
time  that  is  reckoned  at  the  same  instant  at  the  latter 
place.  This  reduction  depends  on  the  difference  of 
longitude  of  the  two  places,  which  for  astronomical 
purposes  is  generally  expressed  in  time;  one  mean  so¬ 
lar  hour  corresponding  to  15°.  An  error  in  the  differ¬ 
ence  of  longitudes,  will  produce  errors  in  the  computed 
longitude  and  latitude  of  the  moon. 

92.  An  observation  of  an  occultation,  at  a  place 
whose  longitude  is  not  correctly  known,  furnishes  one 
of  the  most  accurate  means  of  determining  it.  The  ac¬ 
curacy  will  be  increased,  if  on  the  day  of  occultation, 
the  errors  in  the  tables,  have  been  ascertained  by  ob¬ 
servations  at  a  known  meridian. 

93.  Supposing  the  tables  accurate,  or  that  the  errors 
have  been  ascertained  and  allowed,  the  difference  be¬ 
tween  the  observed,  and  calculated  time  of  the  begin¬ 
ning  or  end  of  an  occultation,  or  of  an  eclipse  of  the 
sun,  at  any  place  whose  latitude  is  accurately  known, 
must  depend  on  errors  in  the  computed  longitude  and 
latitude  of  the  moon,  produced  by  an  error  in  the  lon¬ 
gitude  of  the  place. 

94.  A  small  error  in  the  longitude  of  the  place,  or  which  is  the 
same  in  the  difference  of  time  as  reckoned  at  the  two  meridians, 
will  very  little  affect  the  parallaxes  in  longitude  and  latitude,  as 
is  evident  from  the  formulae  for  computing  these  quantities.  Con¬ 
sequently  the  errors  produced  in  the  apparent  longitude  and  lati¬ 
tude  will  be  sensibly  the  same  as  in  the  true. 

95.  Let  m  and  n  be  the  moon’s  hourly  motions  in  longitude 

21 


154 


ASTRONOMY. 


and  latitude  respectively,  and  x  =  the  error  in  the  difference  of 
meridians.  Then  mx  and  nx  will  be  the  errors  in  the  moon’* 
computed,  apparent  longitude  and  latitude.  Hence,  a  and  c,  be¬ 
ing,  as  before,  the  computed,  apparent  distances  of  the  moon  from 
the  star  in  longitude  and  latitude,  and  s  the  moon’s  apparent  se¬ 
midiameter,  at  the  observed  time  of  the  beginning  or  end,  we 
have, 

(a  -f  mx)  2.  cos2 1  4-  (c  4-  nx)  2  =  s2, 
or,  a2  cos2 1  +  2  mx  cos2 1  +  m2  x2  cos2 1  4-  c2  4-  2  cnx  -f  n2  x2  =  s2 

Now  as  the  longitude  of  the  place  is  supposed  to  be  nearly 
known,  x  must  be  a  small  quantity,  and  the  terms  involving  its 
square  may  be  neglected.  Hence  we  obtain, 

s2  —  c2  —  a 2  cos2 1 

jjj  ,  - _ # 

2  m  cos2 1  +  2  cn 

In  a  similar  manner  we  may  determine  the  longitude  of  a  place, 
from  an  observation  of  an  eclipse  of  the  sun. 

90.  Some  astronomers  think  the  apparent  diameter 
of  the  sun,  obtained  from  observation  and  given  in  the 
solar  tables,  is  t©o  great.  They  infer  this  from  a  com¬ 
parison  of  the  observed  time  of  the  beginning  or  end  of 
a  solar  eclipse,  at  a  known  meridian,  with  the  time  ob¬ 
tained  by  calculation,  after  making  allowance  for  the 
errors  in  the  tables  in  other  respects.  To  account  for 
it,  they  suppose  the  apparent  diameter  of  the  sun  is 
amplified,  by  the  very  lively  impression  so  luminous  an 
object  makes  on  the  organ  of  sight.  This  amplification 
is  called  Irradiation.  Dusejour  thinks  that  in  the  cal¬ 
culation  of  solar  eclipses,  the  semidiameter  of  the  sun, 
as  given  by  the  tables,  ought  to  be  diminished  by  3"£. 
He  also  supposes  the  moon's  atmosphere  inflects  the 
rays  of  light,  so  as  to  produce  an  eflecton  the  beginning 
or  end  of  a  solar  eclipse,  or  of  an  occultation,  equal  to 


CHAPTER  XII*  155 

a  diminution  of  2"  in  the  semidiameter  of  the  moon. 
This  is  called  the  Inflexion  of  the  moon. 

Delambre  is  of  the  opinion  that  the  irradiation  and 
inflexion  are  not  well  established,  and  that  their  ex¬ 
istence  is  very  doubtful. 

CHAPTER  XII. 

Of  the  Planets. 

1.  Hitherto  our  attention  has  been  chiefly  directed  to 
the  Earth,  and  to  those  two  conspicuous  luminaries,  the 
Hun  and  Moon.  We  shall  now  take  some  notice  of 
the  bodies  called  Planets.  These  bodies,  like  the 
moon,  are  observed  to  be  sometimes  on  one  side  of  the 
ecliptic,  and  sometimes  on  the  other.  Their  paths, 
therefore,  cross  the  ecliptic.  Their  apparent  motions 
are  very  irregular;  sometimes  Direct ,  that  is,  from  west 
to  east,  or  according  to  the  order  of  the  signs;  and  some¬ 
times  Retrograde,  or  from  east  to  west.  There  are 
also  times,  at  which  a  planet  appears  to  be  Stationary , 
or  to  have  but  very  little  motiou  for  several  days. 

2.  The  points  in  which  the  path  of  a  planet  cuts  the 
plane  of  the  ecliptic,  are  called  the  Nodes .  The  node 
through  which  the  planet  passes  from  the  south  to  the 
north  side  of  the  ecliptic,  is  called  the  Ascending  node. 
The  other  is  called  the  Descending  node. 

3.  The  Geocentric  place  of  a  body,  is  its  place  as 
seen  from  the  earth.  The  Heliocentric  place,  is  its 
place  as  it  would  be  seen  from  the  sun. 

4  If  a  straight  line  be  conceived  to  be  drawn  from 
the  centre  of  a  planet,  perpendicular  to  the  plane  of  the 
ecliptic,  the  distance  from  the  point,  in  which  it  meets 
the  ecliptic,  to  the  centre  of  the  sun  is  called  the  Cur* 


150 


ASTRONOMY. 


tate  Distance  of  the  planet.  The  point  is  called  the 
Deduced  Place  of  the  planet. 

5.  If  the  reduced  place  of  a  planet,  the  centre  of  the 
sun,  and  centre  of  the  earth,  be  joined  by  three  straight 
lines  they  will  form  a  plane  triangle,  lying  in  the  plane 
of  the  ecliptic.  In  this  triangle,  the  angle  at  the  centre 
of  the  earth  is  called  the  Elongation ;  the  angle  at  the 
centre  of  the  sun,  the  Commutation;  and  the  angle  at 
the  reduced  place  of  the  planet,  the  Annual  Parallax. 

6,  The  sun,  earth,  moon,  and  planets  are  frequently 
designated  by  characters,  as  follows. 


Sun 

O 

Juno 

TT 

4- 

Mercury  - 

$ 

Ceres 

$ 

Venus 

$ 

Pallas 

-  $ 

Earth 

-  0 

Jupiter 

X 

Moon 

3> 

Saturn 

-  h 

Mars 

Vesta 

-  <? 

Uranus 

¥ 

VENUS. 

7-  We  commence  with  Venus,  it  being  the  most  bril¬ 
liant  of  the  planets  and  one  whose  phenomena  are 
easily  observed.  This  planet  always  accompanies  the 
sun,  being  seen  alternately  on  the  east  and  west  side, 
and  never  receding  from  it  more  than  about  45°.  When 
it  is  to  the  east  of  the  sun,  it  is  seen  in  the  evening  aud 
*is  called  the  Evening  Star;  and  when  to  the  west,  it 
is  seen  in  the  morning  and  is  called  the  Morning  Star. 

The  evening  and  morning  star,  called  by  the  ancient 
Greeks,  Hesperus ,  and  Phosphorus ,  were  at  first 
thought  to  be  different  stars.  The  discovery  that  they 
are  the  same,  is  ascribed  to  Pythagoras. 

8.  When  Venus  is  the  evening  star,  and  is  at  its 


CHAPTER  XII. 


1  o7 


greatest  elongation  from  the  sun,  it  appears  through  the 
telescope  to  have  a  semicircle  disc,  like  the  moon  in 
quadratures,  with  its  convexity  turned  to  the  west.* 
From  that  time  as  it  approaches  the  sun,  its  splendour 
increases  for  a  while,  though  the  breadth  of  the  illu¬ 
minated  disc,  diminishes,  like  the  moon  in  the  wane. 
At  the  same  time,  the  diameter,  measured  by  the  dis¬ 
tance  of  the  horns,  increases. 

9.  Yenus  continues  to  approach  the  sun,  till  at  length 
it  becomes  invisible,  in  consequence  of  the  sun’s  supe¬ 
rior  light.  After  some  time  it  appears  on  the  west  side 
and  is  seen  in  the  morning,  before  the  sun  rises. 

10.  Though  Yenus  is  not,  in  general,  visible  at  the 
time  of  its  conjunction  with  the  sun,  it  has  sometimes 
been  seen  as  a  dark  spot,  passing  over  the  sun’s  disc. 
This  phenomenon  is  called  a  Transit  of  Yenus. 
When  Yenus  is  thus  seen  on  the  disc  of  the  sun,  its  ap¬ 
parent  diameter  is  easily  measured,  and  is  found  to  be 
nearly  one  minute. 

11.  As  Yenus  proceeds  to  the  westward  of  the  sun, 
its  disc  is  seen  as  a  crescent,  continually  increasing  at 
the  same  time  the  diameter  is  diminishing.  The  convexi¬ 
ty  is  then  turned  towards  the  east.  When  the  planet  is 
at  its  greatest  western  elongation,  the  disc  is  again  a  se¬ 
micircle.  From  that  time,  as  it  again  approaches  the  sun, 
the  visible  disc,  like  the  moon  after  the  first  quarter, 
approaches  nearer  to  a  circle,  and  just  previous  to  its 
being  lost  in  the  sun’s  rays,  at  the  superior  conjunction 
(10.12.  Note),  the  disc,  does  not  sensibly  differ  from 
a  complete  circle.  Its  diameter  is  then  only  about  10 
seconds. 

12.  From  the  superior  conjunction  the  diameter  of 
Yenus  increases,  but  the  apparent  disc  changes  from  a 

*  In  observing  Venus  with  a  telescope,  it  is  better  to  have  the  object  end, 
partly  covered  in  order  to  diminish  the  light. 


158 


ASTRONOMY. 


full  orb,  till  at  the  greatest  eastern  elongation,  it  again 
becomes  a  semicircle.  The  period,  that  circumscribes 
all  these  changes,  is  the  same  as  the  time  from  one 
conjunction  to  another  of  the  same  kind.  This  period, 
which  is  the  synodic  revolution  (10.12)  of  Venus,  is, 
at  a  mean,  about  584  days. 

13.  The  different  phases  of  Venus  are  readily  ac¬ 
counted  for,  by  supposing  it  to  be  an  opaque,  spherical 
body,  revolving  round  the  sun  from  west  to  east,  at  a 
distance  less  than  that  of  the  earth,  and  shining  by  re¬ 
flecting  the  sun’s  light. 

14  Venus  never  deviates  more  than  a  few  degrees 
from  the  plane  of  the  ecliptic.  By  supposing  it  to 
move  in  a  circular  orbit  in  the  plane  of  the  ecliptic,  wc 
can  easily  obtain  an  approximation  to  its  distance  from 
the  sun,  in  parts  of  the  earth’s  distance.  Calling  the 
earth’s  distance  from  the  sun  1,  let  x  he  the  distance 
of  Venus  from  the  sun.  Then  its  distance  from  the 
earth,  at  inferior  conjunction  will  be  1  —  x;  and  at 
superior,  1  4-  x.  Hence,  (7.13),  1  +  x  :  i  —  x  :  : 
60"  :  10"  :  :  6  :  1,  and  consequently  7x  =  5,  or,  x  = 
.714. 

15.  Since  the  visible  disc  of  Venus  when  at  its  great¬ 
est  elongation,  either  eastern  or  western,  is  a  semicircle, 
it  is  evident  that,  the  annual  parallax  (5)  of  the  planet 
js  then  a  right  angle.  Hence  taking  45°  as  the  great¬ 
est  elongation,  the  distance  of  Venus  from  the  sun, 
found  by  trigonometry  is  .707*  As  the  distance  of 
Venus  from  the  sun,  in  different  positions  is  deter¬ 
mined  to  be  nearly  the  same,  it  appears  the  orbit  is 
nearly  a  circle. 

16.  From  the  synodic  revolution  (12),  the  periodic 
revolution  may  be  determined.  For,  from  the  time 
Venus  is  in  conjunction  with  the  sun,  till  it  is  agaiu  in 


CHAPTER  XII. 


159 


conjunction  of  the  same  kind,  its  angular  motion 
about  the  sun,  must  exceed  the  earth’s  by  a  complete 
revolution,  or  360°.  The  periodic  revolution  of  Ve¬ 
nus  is  224  d.  17  h.  nearly. 


Iff  =  584  =  the  synodic  revolution  in  days,  m  —  59'  8"  = 
the  earth’s  motion  in  one  day,  x  =*=  the  diurnal  motion  of  Venus, 
and  p  5=  the  periodic  time,  we  have, 


or,  x  =  m  + 


tx  =  360°  -J-  fwz, 
360 


and  p  = 


360° 


360°  x  t 
mt  -f  360c 


=  1°.  36', 


—  224  d.  17  h.  nearlv. 


Another  method  of  finding  the  periodic  time  will  be  given  in  a 
succeeding  article. 

17.  Awhile  after  the  greatest  eastern  elongation,  Ve¬ 
nus  comes  nearly  stationary  with  respect  to  the  fixed 
stars,  having  for  a  short  time,  no  sensible  motion  in  lon¬ 
gitude.  After  that  its  motion  becomes  retrograde,  and 
continues  so  till  near  the  greatest  western  elongation,  ' 
when  being  again  a  short  time  stationary,  it  afterwards 
becomes  direct.  The  motion  then  continues  direct 
through  the  remaining  part  of  the  synodic  revolution. 
These  are  necessary  consequences  of  the  respective 
distances  of  the  earth  and  Venus  from  the  sun  and  of 
their  respective  motions  in  their  orbits. 

As  it  will  simplify  the  explanation  and  will  produce 
no  material  error,  we  may  still  suppose  the  orbit  of 
Venus  to  coincide  with  the  plane  of  the  ecliptic.  Let 
S.  Fig  37,  be  the  sun,  BDP/;  the  orbit  of  Venus,  and 
E e  a  p£rt  of  the  earth’s  orbit.  Also,  let  E  and  P,  be 
the  positions  of  the  earth  and  Venus  at  the  time  of  in¬ 
ferior  conjunction,  and  e  and their  places  one  day  af 


160 


ASTRONOMY. 


ter.  Draw  eC  parallel  to  ES.  Now  in  the  triangle 
eSp,  we  know  eS  =  1,  So  =  .71  nearly,  eSp  =  /?SP 
—  eSE  —  1°  36'  —  59'  =  37'*  Whence  the  angle 
S ep  is  found  ==  l°3t'.  Bat  the  angle  SeC  =  ESe 
=  59'.  Since,  therefore,  the  angle  S ep  is  greater 
than  the  angle  SeC,  the  position  of  p  is  to  the  right 
hand  or  west  of  eC,  and  consequently  the  apparent 
motion  of  Venus  is  retrograde,  in  this  part  of  the  orbit. 
It  is  evident,  without  computation,  that  the  motion  of 
Venus  must  be  direct,  in  the  part  of  the  orbit  opposite 
to  the  earth.  It  is  also  plain  that  when  the  motion  is 
changing  from  direct  to  retrograde  or  the  contrary,  the 
planet  must  appear  stationary  for  a  time. 

18.  Admitting  the  truth  of  Kepler’s  third  law  (7-30), 
and  supposing  the  planets  and  earth  to  move  in  circu¬ 
lar  orbits  in  the  plane  of  the  ecliptic,  it  may  be  proved 
by  analytical  investigation,  that  the  apparent  motion  of 
eacli  planet  must  be  retrograde  in  the  part  of  the  orbit 
next  the  earth,  and  direct  in  the  opposite  part.  We 
may  also,  by  such  investigations,  determine  the  Sta¬ 
tionary  points,  that  is,  the  points  at  which  the  planet 
appears  stationary;  and  likewise  the  times  during 
which,  the  motions  of  each  planet  must  appear  retro¬ 
grade.  Now  the  planets  do  not  move  in  the  plane  of 
the  ecliptic,  nor  in  circular  orbits;  but  none  of  them, 
or  at  least,  none  except  Pallas,  deviate  so  far  in  either 
respect,  as  much  to  affect  the  results  obtained  on 
those  suppositions. 

The  computed  durations  of  the  retrograde  motions 
of  the  planets,  arc  found  to  agree  very  nearly  with  the 
durations  obtained  by  observation.  This  near  agree¬ 
ment  forms  a  strong  proof  in  favour  of  the  earth’s  mo¬ 
tion  and  of  the  Copernican  System. 

19.  When  Venus  is  in  the  part  of  the  orbit  opposite 


CHAPTER  XII. 


m 


the  earthy  nearly  the  whole  of  its  enlightened  side  is 
turned  towards  the  earth.  But  on  account  of  its  greater 
distance,  it  does  not  then  afford  so  much  light,  as  when 
in  a  different  part  of  the  orbit.  It  is  found  by  calcu¬ 
lation  that  Yenus  gives  the  most  light  to  the  earth, 
when  being  in  the  inferior  part  of  its  orbit,  its  elonga¬ 
tion  is  39°  43'.  This  takes  place  about  36  days  be¬ 
fore  and  after  inferior  conjunction.  Although  at  those 
times  the  enlightened  disc  is  only  about  one  fourth  of 
the  whole,  the  light  is  so  great  that  Yenus  may  be  dis¬ 
tinctly  seen,  with  the  naked  eye,  in  the  day  time,  even 
when  the  sun  is  shining  in  its  greatest  splendour.  This 
continues  to  be  the  case  for  several  days  at  each  time. 

POSITION  OF  THE  NODES. 

20.  When  a  planet  is  at  either  of  its  nodes,  it  is  in 
the  plane  of  the  ecliptic,  and  consequently  its  latitude 
is  nothing.  Now  from  the  observed  right  ascension 
and  declination  of  a  planet  at  any  time,  its  geocentric 
longitude  and  latitude  may  be  calculated  (6.19).  If 
several  longitudes  and  latitudes  be  thus  obtained,  about 
the  time  the  planet  is  passing  from  one  side  of  the 
ecliptic  to  the  other,  the  exact  time  at  which  its  lati¬ 
tude  is  nothing,  may  be  obtained  by  proportion,  and 
also  its  longitude  at  that  time.  This  longitude  of  the 
planet,  will  evidently  be  the  geocentric  longitude  of  the 
node.  If  similar  observations  and  calculations  be 
made  when  the  planet  returns  to  the  same  node,  we 
shall  again  have  the  geocentric  longitude  of  the  node, 
which  on  account  of  the  different  position  of  the  earth 
in  its  orbit,  will  be  different  from  the  former.  From 
these  two  longitudes,  supposing  the  node  to  have  no 
motion,  its  heliocentric  longitude  may  he  determined. 


162 


ASTRONOMY. 


Let  S,  Fig.  38,  be  the  sun,  N,  the  node,  E,  the  place  of  the 
earth  at  the  time  the  planet  was  found  to  be  in  the  node  from  the 
first  set  of  observations,  and  E'  its  place  at  the  second  time  Also 
let  EQ,  E'Q  and  SQ,  all  parallel  to  each  other,  represent  the  di¬ 
rection  of  the  vernal  equinox.  Put  V  —  SE  =  radius  vector  of 
the  earth  at  the  first  time,  the  mean  radius  vector  being  =  1,  S  = 
QES  —  sun’s  longitude,  G  =  QEN  =  the  geocentric  longitude 
of  the  node,  and  V',  S'  and  G'  the  same  quantities  for  the  second 
time.  Then  if  v  =  SN  =  radius  vector  of  the  planet  when  at 
the  node,  and  N  ==  QSN  ==  the  heliocentric  longitude  of  the  node, 
we  have, 

SEN  =  EQS  —  QEN  =  S—  G, 

SNE  =  QAN  —  QSN  =  QEN  —  QSN  =  G  —  N, 
sin  SNE  :  sin  SEN  :  :  SE  :  SN, 
sin  (G  —  N)  :  sin  (S  —  G)  :  :  V  :  v, 

V.  sin  (S  —  G)  =  v.  sin  (G  —  N). 

In  like  manner,  V'.  sin  (S'  —  G')  =  v.  sin  (G'  —  N). 

Therefore, 

Vsin(S  —  G)  __  sin  (G —  N)  _  sin  G  cos  N — cos  G  sin  N 
V'.  sin  (S' — G')  sin  (G'  —  N)  sin  G'  cos  N  —  cos  G'  sin  N 
sin  G  —  cos  G  tan  N 
sin  G'  —  cos  G'  tan  N 

Hence, 

_  Y.  sin  (S  —  G)  sin  G'  —  V'.  sin  (S'  —  G  )  sin  G 

—  V  sin  (S  —  G)  cos  G'  — V'.  sin  (S'  —  G')  cos  G* 

We  have  also  v  =  — ‘  s*n  ^ 

sin  (G— N) 

21.  From  observations  made  at  distant  periods,  it  is 
found  that  the  nodes  of  the  planets,  have  slow  retro¬ 
grade  motions. 

22.  The  heliocentric  longitude  of  either  node  being 
determined  from  observations  on  the  planet,  when  in 
that  node,  and  the  motion  of  the  node  being  also  as¬ 
certained,  its  heliocentric  lougitude  may  be  found  for 
any  given  time.  When  the  heliocentric  longitudes  of 


CHAPTER  XII. 


163 

the  two  nodes  of  a  planet  are  thus  determined  for  the 
same  time,  they  are  found  to  differ  180°.  Hence  it 
follows,  that  the  line  of  the  nodes,  and  consequently 
the  plane  of  the  orbit,  passes  through  the  centre  of  the 
sun. 


INCLINATION  OF  THE  ORBIT. 

23.  The  place  of  the  node  of  a  planet  being  known, 
the  inclination  of  the  orbit  may  be  determined. 


To  do  this,  find  the  geocentric  longitude  and  latitude  of  the 
planet,  at  the  time  the  longitude  of  the  sun  is  the  same  with  that 
of  the  node.  Let  ENp,  Fig .  39,  be  the  plane  of  the  ecliptic, 
E,  the  earth,  S,  the  sun,  N  the  node,  P,  the  planet,  Pp,  perpen¬ 
dicular  to  the  ecliptic,  and  pD,  perpendicular  to  EN,  the  line  of 
the  nodes.  Then  PDp  is  the  inclination  of  the  orbit.  Put  E  = 
pEN  =.  the  difference  between  the  geocentric  longitude  of  the 
planet  and  the  longitude  of  the  sun,  a  =  the  geocentric  latitude 
of  the  planet,  and  I  =  PDp  =  the  inclination  of  the  orbit.  Then, 


Pp  =  Ep.  tan  a,  and  Dp  =  Ep.  sin  E. 

Hence,tanl=  = 

Dp  sin  E 


PERIODIC  TIME. 

24.  The  interval  from  the  time  the  planet  is  in  one 
of  the  nodes,  till  its  return  to  the  same,  making  allow¬ 
ance  for  the  motion  of  the  node,  gives  the  sidereal 
revolution  of  the  planet.  The  sidereal  revolution  of 
Venus  is  224  d.  16  h.  49  m.  8  sec. 


HELIOCENTRIC  LONGITUDE  AND  LATITUDE,  AND  RA¬ 
DIUS  VECTOR. 

25.  The  place  of  the  node  and  the  inclination  of  the 
orbit,  being  known,  we  may  deduce  the  heliocentric 


m 


ASTRONOMY. 


longitude  and  latitude  of  a  planet,  from  the  geocentric 
longitude  and  latitude,  obtained  from  observation. 


Let  pESN,  Fig.  40,  be  the  plane  of  the  ecliptic,  E,  the 
earth,  S,  the  sun,  P,  the  planet,  N  the  node,  Pp  perpendicular  to 
the  ecliptic,  pD  perpendicular  to  SN,  and  EQ  and  SQ,  the  di¬ 
rection  of  the  vernal  equinox. 

Put  N  —  the  heliocentric  longitude  of  the  node,  S  =  the  sun’s 
longitude,  E  =  the  earth’s  longitude  =  S  -f-  180°,  G  =  the  geo- 
centrical  longitude  of  the  planet,  a  =  the  geocentric  latitude,  L 
==  the  heliocentric  longitude,  l  —  heliocentric  latitude,  I  =  the 
inclination  of  the  orbit,  V  =  the  radius  vector  of  the  earth,  and 
v  =  the  radius  vector  of  the  planet.  Then, 

NSp  =  L  —  N,  SEp  =  G—  S,  ESp  =  E— L,  EpS  =  180° 
—  SEp  —  ESp  =  180°—  G  +  S  —  E  +  L=  180  —  G  +  S 
-^180°  —  S  +  L  =  L  —  G. 


Now,  Ep.  tan  a  =  Pp  =  Sp.  tan.  l3 
tan.  a  Sp  _  sin  SEp  _  sin  (G  —  S) 

tan.  I  Ep  sin  ESp  sin  (E  —  L)’ 

tan.  a  sin  (E  —  L)  =  tan.  I  sin  (G  —  S)  (A). 


Also, 

Dp  =  Sp.  sin  (L  —  N), 

Sp.  tan.  I  —  Pp  =  Dp.  tan.  I  =  Sp.  sin  (L  —  N)  tan.  I, 
tan.  I  =  sin  (L  —  N).  tan  I. 

Hence, 

tan.  a  sin  (E  —  L)  =  sin  (L  —  N)  sin  (G  —  S).  tan.  I, 
or,  tan.  A  sin  [(E  —  N)  —  (L  — -  N)]  =  sin  (L  —  N)  sin 
(G  —  S)  tan  I, 

tan  a  sin  (E  —  N)  cos  (L — N) — tan  acos(E — N)  sin  (L  —  N) 
=  sin  (L  —  N).  sin  (G  —  S)  tan.  I, 

tan  a  sin  (E  —  N)  —  tan  a  cos  (E  —  N)  tan  (L  —  N)  = 
tan  (L  —  N)  sin  (G  —  S)  tan.  I, 

tan  (L — N)  = _ tan  a  sin  (E  — N) _ 

tan  a  cos  (E  —  N)  +  sin  (G  —  S)  tan  I 
Hence,  L  =  (L  —  N)  4-  N,  becomes  knowm 
We  have  also  (A), 


CHAPTER  XII. 


165 


tan.  I  = 


tan  a  sin  (E  —  L) 
sin  (G  —  S) 


From  the  triangle  EpS,  we  have, 

g  _  ESsin  SEp  =  V  sin  (G  —  S) 

^  ~  sin  EpS  sin  (L  —  G) 

„  Sp  V  sin  (G  —  S) 

Hence,  v  =  - =  - - — - k-r* 

cos  PS p  cos.  1.  sin  (L  —  G) 


26.  The  sum  of  the  longitude  of  the  node,  and  of 
the  angle  contained  in  the  order  of  the  signs,  between 
the  right  line,  joining  the  sun  and  node,  and  the  radius 
vector  of  the  planet,  is  called  the  Orbit  Longitude  of 
the  planet.  Thus  the  orbit  longitude  of  the  planet  at 
P,  Fig.  40,  is  QSN  +  NSP. 

If  L'  =  the  orbit  longitude,  then, 

Dp  =  SD.  Ian  (L  —  N),  and  SD.  tan  (L'—  N)  =  DP  =  Py_. 

COS  I 

_  SD.  tan  (L  —  N) 

cos  I 

Hence,  tan  (L'  — N)  =  tan  ^  Z7.  — . 

cos.  I 

LONGITUDE  OF  THE  PERIHELION,  ECCENTRICITY,  AND 
SEMI-TRANSVERSE  AXIS. 

2 7.  Assuming  the  orbit  of  the  planet  to  be  an  ellipse, 
we  may,  from  the  heliocentric,  orbit  longitude,  and 
the  radius  vector,  found  for  three  different  times,  de* 
termine  the  longitude  of  the  perihelion,  the  eccentricity, 
and  the  semi- transverse  axis. 


Let  APD,  Fig.  41,  be  the  orbit,  A  the  perihelion,  and  P,  P', 
and  P  ",  the  three  positions  of  the  planet.  Then  SP,  SP"  and 
SP",  the  three  radius  vectors  are  known,  and  also  from  the  lon¬ 
gitudes,  the  angles  PSP"  and  PSP". 

Put v  =  SP,  V  =  SP',  v"  =  SP",  a;  =  ASP,  *  =  PSP',?  = 


166 


ASTRONOMY. 


PSP",  a  —  semi-transverse  axis,  and  ae  =  the  eccentricity. 
Then,  ( Conic  Sections*) 


—  «•(>-«»)  .  . 

1  +  e.  COS.  X 

«•(!— e2)  . 

1  -f-  e.  cos  (x  4-  6) 

»"=  «•  Q  — «») 

1  4-  e.  cos  (a;  +  <p) 

From  B  and  C, 

v  +  v  e.  cos.  x  =  v'  4-  v'e.  cos  ( x  4-  0) 
vf - V 

or,  e  = - 

v.  cos.  x  —  v'.  cos  ( x  4-  e) 

In  like  manner  from  B  and  D, 

v "  —  v 

e  — 


B 

C 

D 


V . 

COS  X  —  v" 

.  cos  (a;  4-  <p) 

d  v 

" - V  =  ft, 

then  from  E  and  1 

v. 

COS  X  — v'. 

cos.  ( X  +  0) 

v . 

COS.  X — v" 

.  cos.  Kx  4-  <p) 

v . 

COS.  X  —  vf. 

COS.  6  cos.  X  —  v' 

v . 

cos.  X  —  v", 

,  cos.  <p  cos.  X  —  v" 

V  - 

—  v'.  cos.  6 

—  v'.  sin.  6  tan.  x 

v  —  v"0  cos.  <p  —  v".  sin.  <p  tan.  x' 


Hence,  tan.  x  =  ~  .(l"  cos^), 

n  v'.  sin.  6  —  m  v ".  sin.  <p 


The  value  of  x  being  determined,  if  it  be  subtracted  from  the 
orbit  longitude  of  the  planet  in  the  first  position,  the  remainder 
will  be  the  orbit  longitude  of  the  perihelion.  If  L  be  the  ecliptic 
longitude,  and  L'  the  orbit  longitude  of  the  perihelion,  we  have, 
(26), 

tan  (L  —  N)  =  tan  (L/  —  N)  cos  I. 

The  value  of  e,  the  ratio  of  the  eccentricity  to  the  semi-trans¬ 
verse  axis  may  be  found  from  either  of  the  expressions,  E  and  F; 
and  a,  the  semi-axis,  from  either  B,  C,  or  D. 


*  See  Appendix,  article  51. 


CHAPTER  XII. 


167 

28.  When  the  longitude  of  the  perihelion,  the  ec¬ 
centricity,  and  the  semi-transverse  axis,  of  the  orbit  of 
any  planet,  are  determined  from  several  sets  of  obser¬ 
vations,  not  very  remote  from  each  other,  they  are 
found,  respectively,  to  be  very  nearly  the  same.  Hence 
it  appears,  the  assumption,  that  the  orbit  is  an  ellipse, 
is  true,  or  at  least  nearly  so. 

29.  From  observations  made  on  the  different  planets, 
at  remote  periods,  it  is  found  that  the  perihelions  have 
slow  motions.  The  motion  of  the  perihelion  of  Yenus 
is  retrograde.  Those  of  the  other  planets  are  direct. 

The  eccentricities  of  the  orbits  are  also  subject  to 
continued,  but  very  minute  changes.  Some  of  them 
are  at  present  increasing;  others  diminishing. 

The  semi-transverse  axes  of  the  orbits  do  not  change. 
This  fact  was  first  discovered  by  La  Grange,  from  in¬ 
vestigations  in  Physical  Astronomy,  and  it  is  found  to 
he  conformable  to  observation. 

EPOCH  OF  A  PLANET  BEING  AT  THE  PERIHELION  OF  ITS 

ORBIT. 

30.  From  several  observations  of  the  planet  about 
the  time  it  has  the  same  longitude  as  the  perihelion, 
the  correct  time  of  its  being  at  the  perihelion,  may  be 
easily  determined  by  proportion. 

ELEMENTS  OF  THE  ORBIT  OF  A  PLANET. 

31.  The  longitude  of  the  ascending  node  of  the  or¬ 
bit,  the  inclination  of  the  plane  of  the  orbit  to  the  eclip¬ 
tic,  the  mean  motion  of  the  planet  round  the  sun,  the 
mean  distance  of  the  planet  from  the  sun,  or  which  is 
the  same,  the  semi- transverse  axis  of  its  orbit,  the  ec¬ 
centricity,  the  longitude  of  the  perihelion,  and  the  time 


168 


ASTRONOMY. 


when  the  planet  is  in  the  perihelion,  are  called  the 
Elements  of  the  Orbit. 

32.  There  are  various  other  methods,  for  deter¬ 
mining  the  elements  of  the  orbit,  besides  those  which 
have  been  given  in  the  preceding  articles.  Those 
which  are  founded  on  observations  of  the  planet,  when 
in  conjunction,  opposition,  and  in  the  nodes,  are  among 
the  most  accurate. 

The  elements  of  the  orbit,  may  also  be  determined 
with  tolerable  accuracy,  by  certain  methods  of  estima¬ 
tion  and  computation,  without  extending  the  observa¬ 
tions  to  the  time  of  the  planet’s  passage  through  the 
node.  These  methods  were  applied  on  the  discovery 
of  the  new  planets. 

33.  When  the  elements  of  the  planet’s  orbit  have 
been  accurately  determined,  from  a  great  number  of 
observations,  the  equation  of  the  centre  may  be  calcu¬ 
lated,  and  tables  may  be  formed,  which  will  give  the 
heliocentric  longitude,  latitude,  and  the  radius  vector, 
for  any  given  time.  But  most  of  the  planets  are  sen¬ 
sibly  affected  by  the  mutual  attractions,  among  one 
another.  The  perturbations  thus  produced,  have  been 
calculated,  for  several  of  the  planets,  and  form  a  part 
of  complete  tables  of  the  planets. 

GEOCENTRIC  LONGITUDE  AND  LATITUDE. 

34.  From  the  heliocentric  longitude  and  latitude  of 
a  planet,  as  obtainedfrom  the  tables ,  to  find  the  geocen¬ 
tric  longitude  and  latitude. 

Put  p  =*  EpS  =  annual  parallax,  Fig  40,  E  =  SEp  =  the 
elongation,  S  =  ESp  =  the  commutation,  l  =  PS p  =  the  helio¬ 
centric  latitude,  a  ==  PE p  =  the  geocentric  latitude,  V  =  ES  = 


CHAPTER  XII.  189 

radius  vector  of  the  earth,  and  v  =  SP  =  radius  vector  of  the 
planet.  Then  S p  =  v.  cos.  Z. 

Now,  by  trigonometry, 

ES  -f-  Sp  :  ES  —  S p  : :  tan  h  (EpS  +  SEp)  :  tan  £  (EpS  — 
SEp), 

or,  V  -f  v.  cos.  Z :  V  —  v.  cos.  Z :  :  tan  h  (p  4-  E)  :  tan  §  ( p  - —  E). 

Hence,  tan  h  vp  —  E)  =  ^ - — — J  tan  i  ^  _j_  £) 

V  4-  v.  cos.  I 

=  V~  ”•  cosJ.  tan  i  1 180°  —  S) 

V  +  e.  cos.  I  K  ' 


1  — 


V.  cos.  Z 


1  4- 


V.  cos  Z 


tan  (90°  —  h  S). 


Put,  tan.  6  =  _ -™S-— .  Then  (App.  15), 


tan  $  (p  —  E)  =  1 - tau-  tan.  (90  —  b  S) 

U  J  14-  tan.  6  v  J 

=  tan  (45°  —  o').  tan  (90°  —  §  S). 

And  E  =  \  (p  4-  E)  — i(p— E)=  90*  — iS  — 4(p  — E). 

If  ©  =  longitude  of  the  sun,  then 
Geocen.  long,  of  the  Planet  =  ©  4-  E  =  ©  4-  90  —  5  S  — 

HP-  E). 

For  the  geocentric  latitude,  we  have  (25.  A), 

.  .  sin.  E.  tan.  I 

tan.  a  — :  - - — - 

sin  S 

35.  When  a  planet  is  in  conjunction  or  opposition,  the  angles 
of  elongation  and  commutation,  and  consequently  their  sines,  are 
each,  nothing.  In  these  cases  the  geocentric  latitude  can  not  be 
found  by  the  preceding  formula.  It  may  however  be  easily  de¬ 
termined  in  a  different  manner.  Let  E,  Fig.  42,  be  the  earth, 
S  the  sun,  and  P  the  planet,  in  inferior  conjunction.  Then, 


Sp  —  v.  cos.  Z,  and  Pp  —  v.  sin.  1. 

Also  Pp  =  Ep.  tan  a  =  (V  —  v.  cos.  I ).  tan.  a. 
Hence,  (V  —  v.  cos.  l)%  tan  a  =  v.  sin.  Z, 


170 


ASTRONOMY. 


or  tan  a 


v.  sin  l 
V  — v.  cos.  I 


36.  From  the  triangles  SpP  and  EpP,  Fig.  40,  we  easily  ob¬ 
tain  the  distance  of  the  planet  from  the  earth.  Thus, 

EP.  sin.  a  —  Pp  =  v.  sin.  1. 

v.  sin.  I 


or  EP 


sm  a 


37.  Let  «•  =  sun’s  mean  horizontal  parallax,  that  is  the  par- 
rallax  when  the  earth  is  at  its  mean  distance  from  the  sun,  desig¬ 
nated  by  a  unit  or  l,p  =  the  horizontal  parallax  of  the  planet,  in 
a  given  position,  and  R  =  the  radius  of  the  earth.  Then  (5.7), 


1 .  tr  =  R  =  EP.  P) 


» sin.  A 
v.  sin.  I 


The  parallax  of  Venus  when  in  inferior  conjunction  is  about 
30". 


38.  It  has  been  found  from  observations,  that  the  apparent 
semidiaraeter  of  Venus,  when  at  a  distance  from  the  earth,  equal 
to  the  earth’s  mean  distance  from  the  sun,  is  8'  .27.  Hence  if  ^ 
a-  the  apparent  semidiameter  of  Venus,  when  at  a  given  point 
P,  we  have  (7.13), 


8"  27  _  8". 27  sin.  a 
EP  “  v.  sin.  I  ’ 


TRANSIT  OF  VENUS. 

39.  When  Venus  is  in  inferior  conjunction,  if  it  is 
so  near  either  node,  that  its  geocentric  latitude  is  less 
than  the  apparent  semidiameter  of  the  sun,  it  must  pass 
between  the  earth  and  sun,  and  produce  the  pheno¬ 
menon,  called  a  transit  (10).  This  phenomenon  is  of 
great  importance  on  account  of  its  use  in  determining 
the  parallax  of  the  sun,  and  thence  the  real  distances 
and  magnitudes  of  the  bodies  in  the  solar  system. 


CHAPTER  XII.  171 

Its  use  for  this  purpose  was  first  made  known  by  Dr. 
Halley. 

40.  A  synodic  revolution  of  Venus,  being  about  584 
days,  5  synodic  revolutions  will  be  2920  days,  or  8 
years  nearly.  It  follows  therefore,  that  8  years  after 
Venus  has  been  in  inferior  conjunction,  it  is  again  ill 
inferior  conjunction,  nearly  on  the  same  day  in  the 
year,  and  consequently  nearly  in  the  same  part  of  the 
heavens.  Hence  if  it  was  near  the  node,  at  the  former 
conjunction,  it  is  also  near  the  same  node  at  the  latter. 
If  there  was  a  transit  in  the  first  instance,  the  planet 
may  perhaps  be  near  enough  to  the  node,  in  the  se  cond, 
to  occasion  another  transit.  A  more  particular  calcu¬ 
lation  shows  that  whenever  there  is  one  transit  of  Ve¬ 
nus,  there  must,  generally,  be  another  at  the  same 
node,  8  years  before  or  after.  At  the  end  of  a  second 
period  of  8  years,  the  planet  is  too  far  from  the  node, 
at  conjunction,  for  a  transit  to  take  place  When 
there  have  been  two  transits  at  one  node,  the  next  two 
take  place  at  the  other  node,  but  not  till  more  than  a 
century  after  the  former. 

The  last  transits  of  Venus  were  in  the  years  1761 
and  1769.  The  next  will  be  in  1874  and  1882. 

41.  The  time  of  inferior  conjunction  of  Venus,  maybe  found 
from  tables  of  the  sun  and  Venus.  From  the  geocentric  longi¬ 
tude  and  latitude  of  Venus,  calculated  for  the  time  of  conjunction, 
and  also  for  a  time,  an  hour  before  or  after  conjunction,  the  geo¬ 
centric  hourly  motions  of  Venus  in  longitude  and  latitude  will  be 
known.  The  geocentric  hourly  motion  in  longitude,  being  re¬ 
trograde  must  be  added  to  the  sun’s  hourly  motion,  to  obtain  the 
relative  hourly  motion,  which  will  also  be  retrograde.  From  the 
hourly  motion  in  latitude,  and  the  relative  hourly  motion  in  longi¬ 
tude,  the  inclination  of  the  relative  orbit  may  be  found,  in  the 
same  manner,  as  the  inclination  of  the  moon’s  relative  orbit,  in 


472 


ASTRONOMY. 


eclipses  of  the  moon.  The  apparent  semidiameter  of  Venus  may 
be  found  by  the  formula  in  a  preceding  article  (38). 

42.  Let  LMNQ,  Fig.  43,  represent  the  disc  of  the  sun,  C  its 
centre,  AB  a  part  of  the  ecliptic,  the  order  of  the  signs,  being 
from  A  to  B,  CL  a  circle  of  latitude,  and  vw  the  relative  orbit  of 
Venus.  The  apparent  motion  of  Venus  being  retrograde,  will  be 
in  the  direction,  from  v  to  w. 

The  position  of  the  relative  orbit  is  adapted  to  the  transit  of 
1769,  at  which  time  the  latitude  of  Venus  was  north,  decreasing. 

With  the  centre  C  and  a  radius  equal  to  the  sum  of  the  semi¬ 
diameters  of  the  sun  and  Venus,  let  arcs  be  described,  cutting  vw, 
in  v  and  tr,  and  with  a  radius  equal  to  the  difference  of  the  semi¬ 
diameters,  let  other  arcs  be  described,  cutting  vw  in  a  and  e.  The 
situations  of  Venus  with  respect  to  the  sun,  when  at  the  points  v, 
ct,  e,  and  to,  are  respectively  called  the  First  External  Contact , 
First  Internal  Contact ,  Second  Internal  Contact ,  and  Second  Ex¬ 
ternal  Contact . 

43.  The  times  of  the  different  contacts  as  seen  from  the  cen¬ 
tre  of  the  earth,  may  be  calculated  in  the  same  manner  as  the 
beginning  and  end  of  an  eclipse  of  the  moon,  using  the  geocentric 
latitude  of  Venus,  instead  of  the  moon’s  latitude,  the  sum  or  dif¬ 
ference  of  the  semidiameters  of  the  sun  and  Venus,  instead  of  the 
sum  of  those  of  the  earth’s  shadow  and  moon,  the  geocentric  hourly 
motion  of  Venus  in  latitude,  instead  of  the  moon’s  hourly  motion 
in  latitude,  and  the  relative  hourly  motion  of  Venus  in  longitude, 
instead  of  that  of  the  moon. 

PARALLAXES  OF  THE  SUN  AND  VENUS. 

44.  Let  E,  Fig.  44,  be  the  earth,  V,  Venus,  and 
acdh'b,  the  disc  of  the  sun.  Also  let  AGE  be  the  path 
of  Venus  over  the  sun’s  disc  as  seen  from  E,  the  earth’s 
centre,  the  latitude  of  Venus  being  north.  Then  to  a 
spectator  at  1  towards  the  north  pole,  the  path  will  be 
all/;,  which  being  nearer  the  centre  is  greater  than  the 
former.  Consequently  the  duration  of  the  transit  as 
seen  from  I,  is  longer,  than  as  seen  from  the  centre  of 


CHAPTER  XI r. 


173 


the  earth.  To  a  spectator  at  K,  towards  the  south  pole, 
the  path  will  he  a'F//,  which  is  less  than  AB.  Hence 
the  duration  is  shorter  than  as  seen  from  the  earth’s 
centre.  The  difference  between  the  durations  as  seen 
from  I  and  K,  depends  on  the  parallaxes  of  Venus 
and  the  sun,  or  rather  on  the  difference  between  the 
parallax  of  Venus  and  that  of  the  sun.  This  differ¬ 
ence  is  called  the  Relative  Parallax  of  Venus. 

From  the  observed  durations  of  the  transit  at  the  two 
places,  the  relative  parallax,  and  thence,  the  parallax 
of  the  sun,  may  be  determined. 

45.  Let  a'  Fig  43,  be  the  place  of  the  centre  of  Venus,  at  the  1st 
internal  contact  as  seen  from  a  place  A  in  north  latitude,  where 
the  commencement  is  accelerated,  and  end  retarded  by  the  effects 
of  parallax,  and  let  aS  and  «'R  be  each  perpendicular  to  AB. 
Put, 

T  =  time  of  1st  internal  contact  for  the  earth’s  centre, 

0  =  the  same  for  the  place  A, 
t  =  T —  0, 

E  —  CS  =  the  elongation  of  Venus  at  the  time  T, 

G  =  aS  =  the  geocentric  latitude  at  the  same  time, 
m  =  relative  hourly  motion  of  Venus  in  longitude, 
r  -  -  hourly  motion  of  Venus  in  latitude, 

P  =  relative  parallax  of  Venus, 
n  —  relative  parallax  in  longitude  at  the  time  0, 

5r  =  relative  parallax  in  latitude  at  the  same  time. 

Then  CR  =  E  4-  mt  +  n%  and  a' R  =  G  -f  rt  +  *■*, 

CR2  +  a'R2  =  rt'C2  ==  aC2  =  CS2  4-  aS2, 
or,  (E  4-  mt  +  n)2  +  (G  +  rt  4-  *)2  =  E2  4-  G2, 

*  The  unknown  quantities  n  and  vr,  are  applied  with  affirmative  signs,  al¬ 
though,  one  of  them  at  least,  in  the  present  case,  must  be  negative.  This 
produces  no  error  in  the  investigation.  In  the  numerical  computation,  these 
quantities  must  be  used  -with  the  signs,  which  they  will  be  determined  to 
have  in  particular  cases. 


174 


ASTRONOMY. 


2E mt  +  2En  4-  2mtn  4-  m2t 2  +  n2  +  2G rt  +  20*  +  2rt *  4. 
r2t2  +  Jr2  =  0, 

( m 2  -f  r2).  t2  -f  2.  (Em  -f  mu  4-  Gr  4-  rir).  t  =  —  2.  (En  4. 

Gt  4-  §  n2  4-  i  5T2) 

^2  (Em  4-  mil  4-  Gr  4-  r?r> 

,  ??i2  4-  r2 

_ 2.  (En  4-  Gx  4.  I  n2  4-  £  ,r2) 

m2  4-  r2 

_  Em  4-  win  4-  Gr  4-  r* 
m2  4-  r2 

and  6  =  2.(En  +,G^  +  in-  +  ^)  we  h 

m2  +  r2 

t2  4.  2at  =  —  b 
t 2  4-  2at  4-  a2  =  a2  —  b 

'  +  “  =  ^  ^~6>  =  a-Ta~8ir>  _&C- 
i  =  ~ A  —  &c 

2a  8a3 


7  2 

But  _ ,  and  all  the  following  terms  of  the  value  of  t ,  are  ex- 

8  a3 

tremely  small  and  may  be  neglected.  Hence, 

t  _b _ En4GT.^nJ4§5r2 

2a  Em  4-  mn  4.  Gr  4-  r* ' 

_  En  4-  Gw  4.  ;§  n2  _j_  §  <r2 


(E  4-  n).  m  4  (G  4  *').  r 
Put  n  =  aP,  and  *  =  vP,  then, 

. _ Etc  4-  Gv  +  j  un  4  j  v*  p  , 

(E  4-  n).  m  4-  (G  4-  *>'  ’  * 

p _ q  ,  i  ~-g _ Etc  -j-  Gv  4~  2  air  4-  i  ^7r  p 

(E  4-  n).  m  4  (G  4-  *).  r 


46.  Let  T'  be  the  time  of  the  2d  internal  contact  for  the  earth’s 
centre,  and  0',  for  the  place  A.  And  instead  of  E,  G,  m,  r,  n,  *■, 
u  and  v,  let  E',  G',  m',  r',  rr,  u'  and  v'  be  the  values  of  the  cor¬ 
responding  quantities  at  the  2d  internal  contact.  As  the  latitude 
is  decreasing  and  the  2d  internal  contact  for  the  place  A,  is  later 
than  for  the  earth’s  centre,  the  value  of  r  must  be  taken,  negative, 
in  obtaining  the  latitude  at  the  latter  time,  from  that,  at  the 


CHAPTER  XII.  175 

former.  Attending  to  these  changes,  we  have  in  like  manner  as 
before, 

T,  =  6  ,  E V  +  GV  +  In'ii'  +ji>V  p 
=  (E'  4-  n').  m'  —  [G'  +  **  )•  r' 

Put  a  -  E..+  G.4  inn,-  jv*  and 
(E  +  n).  m  +  (G  +  sr).  r 
p,  _  E 'u'  4-  G  v ’  4-  i  if  n'  4-  l  v'v' _ 

~~  (E'  4  ir;.  »f  —  4-  *■').  / 

Then  T  =  0  —  /3P,  and  T'  =  *'  4.  /8'P, 

T'  —  T  =  6’  —  6  4-  (/3  4-  /3').  P. 

If  now  d  —  V  —  0,  and  s  =  /3  4-  /3',  we  hare, 

T'  —  T  =  d  4-  sP. 

47.  If,  for  some  other  place  in  a  latitude  considerably  different 
from  that  of  A,  d1  and  s'  be  the  values  of  the  expressions  desig¬ 
nated  by  d  and  s,  we  shall  have  in  like  manner, 

T'  —  T  =  d'  4-  s'P. 

Hence,  d  4-  sP  =  d!  4-  s'P, 
sP  —  s'P  =  d1  —  d, 

p  _  d'  —  d 

s  —  s'* 

48.  In  this  expression  for  the  value  of  P,  the  two  quantities  d 
and  d'  are  known  from  observation.  The  quantities  s  and  s'  de¬ 
pend  on  the  values  of  /3  and  ja'  for  the  two  different  places  of  ob¬ 
servation.  In  the  expression  represented  by  fl,  the  values  of  E, 
G,  m  and  r,  are  known.  To  obtain  u  and  v,  let  L  —  the  geo¬ 
centric  longitude  of  Venus,  and  A  =  its  distance  from  the  north 
pole  of  the  ecliptic.  Then  ( 10.52. C.  and  54.G), 

sin  n  =  P  sin,  ft  sin  (L  n  4-  n> 
sin.  A 

sin  sr  =  sin  P  cos  h  sin  (A  4-  *-) 

___  sinP  sin  h  cos  (L  —  n  \  n)  cos  (A  4-  «•) 
cos  \  n 

Or  because  n  and  are  very  small, 


176 


ASTRONOMY. 


sm  n  = 


sin  P  sin  h  sin  L  —  n) 


sin  A 


(Q) 


in  %  =  sin  P  ^ cos  h  sin  A  —  sin  h  cos  A  cos  L  —  n ^  (R) 

sirfl  sin  h  sin  L  —  n) 


Hence  45  ,  u  = 


sin  P 


sin  A 


%  sm  7r  i  •  *  •  /  .  n 

v—  —  =  -  =  cos  h  sin  A  —  sin /i  cos  A  cos  L  —  nj. 

P  sin  P  ^  J 


The  other  two  quantities  n  and  tt,  contained  in  the  expression 
represented  by  /3,  depend  on  P,  the  quantity  sought.  But  as  n  and 
9r  are  very  small,  it  is  evident  from  examination  of  the  expression, 
that  we  obtain  a  near  value  of  /3,  if  the  terms,  in  which  they  en¬ 
ter,  are  omitted.  Hence  by  taking, 


E  u  Gv 
Em  -  Gr 


and  /3'  = 


EV  4-  GV 

E 'm'  —  G'r'’ 


we  obtain  a  very  near  approximate  value  of  P.  Then,  taking  n 
3=  wP,  v  vP,  IT  u' P',  and  «•'  =  v'P,  and  again  calculating 
the  values  of  /3  and  /3',  we  obtain  the  correct  value  of  P. 

49.  If  sr'  =  the  sun’s  horozontal  parallax  at  the  time  of  the 
transit,  sr,  being  the  mean  parallax,  and  V  the  radius  vector,  we 


have  *r'  =  — . 

At  the  time  of  a  transit,  the  latitude  of  Venus  is  so  small  that 

we  may  consider  its  sine  as  equal  to  its  tangent,  and  its  cosine 
=  1,  Hence  ^35), 


sine  a  =  ^_-Sin‘--,  and  (37% 

V  —  v 

sin  A  _  zr 

v.  sin  l  V  —  v 

Therefore  P  —  « _ «r'  =  *L_  _ —  ...  ^ 

ineieiore  r  _  p  *  y  y  -  y2  _  yj) 

V2  —  V» 

Hence,  w  =  P.  - — 

v 


50.  The  transit  of  1769  was  observed  at  Ward  bus, 
a  small  island  on  tlie  north  coast  of  Europe;  and  at 


CHAPTER  XII. 


Otaheite  in  the  South  Sea,  and  the  duration  was  found 
to  he  longer  at  the  former  place,  than  at  the  latter,  by 
23  m.  10  sec.  The  sun’s  mean  horizontal  parallax, 
determined  from  the  observations,  made  at  those  places, 
is  8".7.  From  observations  made  at  other  places,  re¬ 
sults  a  little  different  were  obtained.  By  taking  the 
mean  of  the  results  deduced  from  the  most  accurate  ob¬ 
servations,  astronomers  have  fixed  the  parallax  at  8". 6 
or  8".7;  some  adopting  one  number,  and  some  the 
other. 

51.  Taking  the  sun’s  parallax  8".7,  the  earth’s 
mean  distance  from  the  sun  is  23708  semidiameters  of 
the  earth  (5-8),  or  94,000,000  English  miles,  nearly. 
Thence  from  the  sidereal  revolutions  of  the  earth  and 
Venus,  the  mean  distance  of  Venus  from  the  sun,  found 
by  Kepler’s  third  law  (7-30),  is  68  millions  of  miles. 
From  the  observed  diameter  of  Venus,  when  at  a 
known  distance  from  the  earth,  its  real  diameter  is 
easily  found.  It  is  about  7600  miles. 

62.  It  is  found  that  Venus  revolves  on  its  axis,  from 
west  to  east,  in  23  h.  21  m.,  and  that  its  axis  is  in¬ 
clined  to  the  ecliptic  in  an  angle  of  about  15°. 

MERCURY. 

53.  Mercury,  like  Venus,  always  accompanies  the 
sun.  Its  greatest  elongation  is  about  23°.  The  phe¬ 
nomena  of  Mercury  correspond  in  almost  every  part, 
with  those  of  Venus,  only  that  it  is  farther  from  the 
earth  and  nearer  to  the  sun,  and  consequently  more  dif¬ 
ficult  to  be  observed.  Its  greatest  and  least  apparent 
diameters  are  11". 2  and  5".  It  can  only  be  seen,  by 
the  naked  eye,  when  in  the  most  favourable  positions. 

Mercury  makes  a  sidereal  revolution  round  the  sun, 

24 


178 


ASTRONOMY. 


in  about  88  clays,  at  a  mean  distance  of  37  millions  of 
miles.  Its  diameter  is  a  little  more  than  3,000  miles. 

On  account  of  the  proximity  of  Mercury  to  the  sun, 
it  is  difficult  to  determine  whether  it  revolves  on  its 
axis.  Shroeter  thinks  he  has  ascertained,  that  it  makes 
a  revolution,  like  the  earth  and  Venus,  from  west  to 
east,  in  24  h.  5  m.;  and  that  its  axis  makes  hut  a  small 
angle  with  the  ecliptic. 

54.  When  at  the  time  of  inferior  conjunction,  Mer¬ 
cury  is  in  either  node,  or  very  near  to  it,  a  transit  of 
Mercury  takes  place.  Transits  of  Mercury  occur 

.  much  more  frequently  than  those  of  Venus.  The 
next  five  will  take  place  in  the  years,  1822, 1832,  1835, 
1845,  and  1848.  Of  these,  the  last  four  will  he  visi¬ 
ble  in  the  United  States. 

The  calculation  of  a  transit  of  Mercury  is  altogether 
similar  to  one  of  Venus. 

55.  Mercury  and  Venus  are  called  Inferior  planets, 
because  their  orbits  are  included  within  the  earth’s. 
The  others  are  called  Superior,  because  their  orbits 
are  without  the  earth’s. 

MARS. 

56.  Mars  and  all  the  other  superior  planets,  differ 
from  Mercury  and  Venus,  in  beiug  seen  in  opposition 
as  Well  as  in  conjunction.  The  disc  of  Mars  does 
not  present  the  phases  of  the  two  inferior  planets,  hut 
it  is  observed,  in  particular  situations,  to  deviate  very 
sensibly  from  a  circle.  The  apparent  diameter  of 
Mars,  undergoes  considerable  change.  When  great¬ 
est  it  is  17">  and  when  least,  3".5. 

The  sidereal  revolution  of  Mars  is  nearly  687  days, 
and  its  mean  distance  from  the  sun  is  143  millions  of 
tniles.  Its  diameter  is  about  4000  miles. 


CHAPTER  XII. 


179 


Mars  revolves  on  its  axis  from  west  to  east  in 
24  h.  39  m.,  and  its  axis  is  inclined  to  the  ecliptic  in  an 
angle  of  59°  42'.  Its  polar  diameter  is  less  than  the 
equatorial.  According  to  the  measures  of  Arago,  these 
diameters  are  to  each  other  in  the  ratio  of  189  to  194. 

JUPITER  AND  ITS  SATELLITES. 

57.  Jupiter  is  the  most  brilliant  of  the  planets,  ex¬ 
cept  Venus.  Its  apparent  diameter  when  greatest  is 
44".5,  and  when  least,  30". 

The  sidereal  revolution  of  Jupiter  is  about  4333 
days,  or  nearly  12  years,  and  its  mean  distance  from 
the  sun  is  nearly  490  millions  of  miles.  The  diameter 
of  Jupiter  is  89,000  miles,  which  is  more  than  1 1  times 
the  diameter  of  the  earth.  The  magnitude  of  Jupiter 
is  therefore  more  than  1300  times  that  of  the  earth. 

Jupiter  revolves  from  west  to  east,  on  an  axis  nearly 
perpendicular  to  the  ecliptic,  in  9  h.  56  m.  Its  polar 
diameter  is  to  its  equatorial  diameter,  in  the  ratio  of 
167  to  177. 

58.  W  hen  Jupiter  is  examined  with  a  good  telescope, 
its  disc  is  observed  to  be  crossed  near  the  centre  by 
several  obscure  spaces  which  are  nearly  parallel  to 
each  other,  and  to  the  plane  of  the  equator.  These 
are  called  the  Belts  of  Jupiter. 

59.  When  Jupiter  is  viewed  with  a  telescope,  even 
of  moderate  power,  it  is  seen  accompanied  by  four 
small  stars,  nearly  in  a  straight  line  parallel  to  the 
ecliptic.  These  always  accompany  the  planet,  and  are 
called  its  Satellites .  They  are  continually  changing 
their  positions  with  respect  to  one  another,  and  to  the 
planet,  being  sometimes  all  to  the  right,  and  some¬ 
times  all  to  the  left;  but  more  frequently  some  on  each 
side.  The  greatest  distances  to  which  they  recede 


180 


ASTRONOMY. 


from  the  planet  on  each  side,  are  different  for  the  dif¬ 
ferent  satellites,  and  they  are  thus  distinguished;  that 
being  called  the  First  satellite,  which  recedes  to  the 
least  distance;  that  the  Second ,  which  recedes  to  the 
next  greater  distance,  and  so  on.  The  satellites  of 
Jupiter  were  discovered  by  Galileo  in  1610. 

60.  Sometimes  a  satellite  is  observed  to  pass  between 
the  sun  and  Jupiter,  and  to  cast  a  shadow  which  de 
scribes  a  chord  across  the  disc.  This  produces  an 
eclipse  of  the  sun,  to  Jupiter,  analogous  to  those  which 
the  moon  produces  on  the  earth.  It  follows  that  Jupi¬ 
ter  and  its  satellites  are  opaque  bodies,  which  shine 
by  reflecting  the  sun's  light. 

Jupiter  being  an  opaque  and  nearly  spherical  body, 
must  project  a  conical  shadow  in  a  direction  opposite 
to  the  sun.  When  either  of  the  satellites  enters  this 
shadow,  it  must  suffer  an  eclipse  and  consequently  be¬ 
come  invisible.  Observations  show  that  this  is  the  case. 
The  satellites  are  frequently  seen,  even  when  con¬ 
siderably  distant  from  the  planet,  to  grow  faint,  and  in 
a  little  time,  entirely  to  disappear.  The  third  and 
fourth  satellites  are  sometimes  observed,  after  having 
been  eclipsed,  again  to  become  visible  on  the  same  side 
of  the  disc.  These  phenomena  indicate  that  the  satel¬ 
lites  of  Jupiter  are  little  moons  which  revolve  round 
that  planet,  in  like  manner  as  the  moon  does  round  the 
earth. 

61.  The  satellites  are  sometimes  on  the  opposite  side 
of  Jupiter,  from  the  earth,  and  consequently  become  in¬ 
visible.  Sometimes  they  are  between  the  earth  and 
Jupiter,  in  which  case  they  are  not  easily  distinguish¬ 
ed  from  the  planet  itself. 

When  a  satellite  is  invisible  in  consequence  of  en¬ 
tering  into  the  shadow  of  Jupiter,  the  phenomenon  is 
called  an  Eclipse  of  the  satellite. 


CHAPTER  XII, 


ist 


63.  Careful  and  repeated  observations,  show  that 
the  motions  of  the  satellites,  are  from  west  to  east,  in 
orbits  nearly  circular,  and  making  small  angles  with 
the  plane  of  Jupiter’s  orbit.  Observations  on  the 
eclipses  of  the  satellites  make  known  their  synodic  re¬ 
volutions,  from  which  their  sidereal  revolutions  are 
easily  deduced.  From  measurements  of  the  greatest  ap¬ 
parent  distances  of  the  satellites  from  the  planet,  their 
real  distances  are  determined. 

63.  A  comparison  of  the  mean  distances  of  the  sa¬ 
tellites,  with  their  sidereal  revolutions,  proves  that 
Kepler’s  third  law  with  respect  to  the  planets  applies 
also  to  the  satellites  of  Jupiter.  The  squares  of  their 
sidereal  revolutions  are  as  the  cubes  of  their  mean  dis¬ 
tances  from  the  planet. 

The  planets  Saturn  and  Uranus  are  also  attended 
by  satellites,  and  the  same  law  has  place  with  them. 

6-1.  The  crbits  of  the  third  and  fourth  satellites  are 
elliptical.  Those  of  the  other  two,  have  not  been  as¬ 
certained  to  differ  sensibly  from  circles. 

6J.  The  mutual  attractions  of  the  satellites  on  one 
another,  produce  inequalities  in  their  motions,  which 
must  be  taken  into  view,  wlfm  it  is  designed  to  deter¬ 
mine  from  calculation,  their  positions,  at  any  given 
time,  with  accuracy.  In  investigating  this  subject, 
La  Place  discovered  two  very  remarkable  conditions, 
that  connect  the  mean  motions  of  the  first  three 
satellites. 

He  found,  That  the  mean  motion  of  the  first  satellite , 
added  to  twice  the  mean  motion  of  the  third ,  is  exactly 
equal  to  three  times  the  mean  motion  of  the  second . 

He  also  found,  That  the  mean  longitude  of  the  first 
satellite ,  less  three  times  that  of  the  second ,  more  twice 


ASTRONOMY". 


182 

that  of  the  third ,  must  always  he  equal  to  180°.  It 
follows  from  this  circumstance  that  the  longitudes  of 
these  three  satellites  can  never  be  the  same  at  the  same 
time,  and  consequently  that  they  can  never  be  all 
eclipsed  at  once. 

66.  The  satellites  of  Jupiter  undergo  periodical 
changes  in  brightness.  From  very  attentive  observa¬ 
tions  of  these  changes,  Dr.  Herschel  infers  that  each 
satellite  revolves  on  its  own  axis  in  the  same  time 
that  it  makes  a  sidereal  revolution  round  the  planet. 

67-  Observations  on  the  eclipses  of  Jupiter’s  satel¬ 
lites,  have  led  to  the  discovery  of  a  very  important  fact; 
which  is,  that  the  Transmission  of  light  is  Successive. 

When  Jupiter  is  in  opposition,  the  eclipses  of  the 
satellites,  happen  earlier,  than  they  ought  to  do,  ac¬ 
cording  to  the  known  durations  of  their  revolutions, 
and  on  the  supposition  that  the  transmission  of  light 
is  instantaneous.  On  the  contrary,  when  Jupiter  is 
near  conjunction,  they  happen  later  than  they  ought  to 
do  on  the  preceding  supposition.  The  variations  are 
the  same  for  all  the  satellites,  and  are  found  evidently 
to  he  connected  with  the  distance  of  Jupiter  from  the 
earth;  the  eclipses  happening  later  as  the  distance  is 
greater.  These  circumstances  are  easily  explained 
and  the  amount  of  the  retardation,  accurately  account¬ 
ed  for,  by  allowing  light  to  occupy  16  m.  26  sec.  in 
traversing  with  a  uniform  motion,  a  distance  equal  to 
the  transverse  axis  of  the  earth’s  orbit.  The  discovery 
of  the  successive  transmission  of  light  was  made  by 
ilcemer  a  Danish  astronomer,  in  the  year  I67J. 

68.  Since  light  is  16  m.  26  sec.  in  passing  over  a 
distance  equal  to  the  diameter  of  the  earth’s  orbit,  it 
must  be  8  m.  13  sec.  in  passing  from  the  sun  to  the 
earth,  when  these  bodies  are  at  their  mean  distance. 


CHAPTER  XII. 


183 


Its  velocity  is  therefore  180,000  miles  per  second, 
which  is  greater  than  any  other,  with  which  we  are 
acquainted. 

69.  The  eclipses  of  Jupiter’s  satellites  furnish  a 
simple  means  of  determining  the  longitudes  of  places 
on  land,  with  considerable  accuracy. 

The  tables  for  calculating  these  eclipses,  constructed 
by  Delambre,  and  founded  on  the  theory  of  La  Place, 
give  the  times  of  beginning  or  end  of  the  eclipses,  with 
very  little  error.  These  times  are  calculated  and  in¬ 
serted  in  the  Nautical  Almanac,  for  the  meridian  of 
Greenwich,  and  in  the  Connaissance  Be  Terns ,  for  the 
meridian  of  Paris. 

As  a  satellite  really  loses  its  light  by  entering  into 
the  shadow  of  Jupiter,  the  commencement  of  an 
eclipse  must  be  seen  at  the  same  instant  by  all  ob¬ 
servers,  however  distant  from  one  another*.  If,  there¬ 
fore  an  eclipse  of  one  of  the  satellites,  be  observed  at 
a  place  whose  longitude  is  required,  the  difference  be¬ 
tween  the  observed  time,  and  the  time  computed  for 
the  meridian  of  Greenwich,  will  give  the  difference  of 
meridians,  supposing  the  tables  to  be  accurate. 

This  method  of  finding  the  longitude  can  not  be  em¬ 
ployed  at  sea,  because  the  motion  of  the  vessel,  pre¬ 
vents  the  use  of  telescopes  of  sufficient  power,  for  ob¬ 
serving  the  eclipses. 

SATURN  WITH  ITS  SATELLITES,  AND  RING. 

70.  Saturn  revolves  round  the  snn  in  about  10758 

*  This  supposes  the  telescopes  used  by  the  observers  to  be  of  equal  good¬ 
ness.  For,  since  the  diminution  of  light  is  gradual  (60),  two  observers,  by 
the  side  of  each  other,  but  using  telescopes  of  differnt  power,  will  not  lose 
sight  of  the  satellite  at  the  same  instant.  The  observation  also  depends  on 
the  state  of  the  air,  and  in  some  measure  on  the  eye  of  the  observer. 


484 


ASTRONOMY. 


(lays,  or  nearly  29  \  years,  at  the  distance  of  900  mil¬ 
lions  of  miles.  Its  diameter  is  79,000  miles.  The 
greatest  and  least  apparent  diameters  of  Saturn,  are 
20". 1  and  16".3. 

Saturn  revolves  on  its  axis  from  west  to  east,  in 
40  h.  16  m.  Its  axis  is  inclined  to  the  ecliptic  in  an 
angle  of  about  60°.  The  polar  diameter  is  to  the 
equatorial,  in  the  ratio  of  40  to  44. 

71,  The  planet  Saturn  is  distinguished  from  all  the 
other  planets,  in  being  surrounded  by  a  broad,  thin 
ring,  which  is  entirely  detached  from  the  body  of  the 
planet.  It  is  ascertained  to  be  opaque  and  to  shine  by 
reflecting  the  sun’s  light.  This  ring  was  discovered 
by  Huygens,  and  is  discernible,  when  in  favourable 
positions,  with  telescopes  of  small  power. 

The  plane  of  the  ring  is  inclined  to  the  ecliptic  in  an 
angle  of  31°  21'.  Consequently,  the  face  of  the  ring 
can  never  be  turned  directly  towards  the  earth.  It  is 
generally  seen  under  the  form  of  an  eccentric  ellipse. 
The  ring  becomes  invisible  when  the  enlightened 
face  is  turned  from  the  earth.  On  account  of  its  little 
thickness,  it  is  also  invisible  in  two  other  cases.  These 
are,  when  the  plane  of  the  ring,  produced,  passes  through 
the  centre  of  the  earth,  and  when  it  passes  through  the 
centre  of  the  sun. 

The  ring  revolves  round  an  axis,  perpendicular  to 
its  plane,  and  passing  through  the  centre  of  Saturn,  in 
40  h.  29  m. 

Observations  with  telescopes  of  high  power,  show 
that  the  ring  of  Saturn,  really  consists  of  two  concen¬ 
tric  rings,  entirely  separate  from  each  other.  The 
breadth  of  the  interior  ring  is  20,000  miles;  of  the  ex¬ 
terior  7000,  and  of  the  space  between  them  2800.  The 


CHAPTER  XII.  185 

distance  from  the  centre  of  the  planet  to  the  inside  of 
the  interior  ring  is  73,000  miles. 

72.  Saturn  is  accompanied  by  seven  satellites,  which 
move  round  it  from  west  to  east  in  orbits  that  are  nearly 
circular.  The  orbits  of  the  first  six,  nearly  coincide 
with  the  plane  of  the  ring;  that  of  the  seventh,  makes 
a  less  angle  with  the  ecliptic. 

URANUS  AND  ITS  SATELLITES. 

73.  The  planet  Uranus  was  discovered  by  Dr. 
Herschel  in  the  year  1781.  It  revolves  round  the  sun 
in  30689  days,  or  a  little  more  than  84  years,  at  the 
distance  of  1800  millions  of  miles.  Its  diameter  is 
35000  miles.  The  greatest  and  least  apparent  diame¬ 
ters  are  4".l  and  3". 7-  The  distance  of  Uranus  is  so 
great  that  its  revolution  on  its  axis  has  not  been  as¬ 
certained. 

According  to  the  observations  of  Herschel,  Uranus 
is  accompanied  by  six  satellites,  which  revolve  in 
orbits  nearly  perpendicular  to  the  plane  of  the  ecliptic. 

VESTA,  JUNO,  CERES  AND  PALLAS. 

74.  These  four  planets,  although  less  distant  than 
several  of  the  others,  are  so  extremely  small  that  they 
can  only  be  seen  with  telescopes  of  considerable 
power.  Ceres  was  discovered  by  Piazzi  on  the  first 
day  of  the  present  century;  Pallas,  by  Olbers  in  1802; 
Juno,  by  Harding  in  1803;  and  Yesta,  by  Olbers  in 
1807.  They  revolve  from  west  to  east,  in  orbits  not 
very  different  in  extent,  and  contained  between  those 
of  Mars  and  Jupiter.  The  orbit  of  Pallas  differs 
from  those  of  all  the  other  planets  in  the  greatness  of 


186 


ASTRONOMY, 


the  angle,  it  makes  with  the  ecliptic.  This  angle  is 
nearly  35°. 

75.  The  following  tallies  contain  the  elements  of 
the  orbits  of  the  planets  and  the  periodic  revolutions 
of  the  satellites.  The  elements  of  the  four  new  planets 
are  to  be  considered  only  as  approximations. 

Sidereal  Mevolutions  of  the  Planets, 


Days. 


Mercury 

- 

87.969258 

Venus 

- 

224.700t87 

The  Earth 

- 

365.256384 

Mars 

- 

686.979646 

Vesta 

- 

1326.930 

Juno  - 

- 

1594.023 

Ceres 

- 

1681.370 

Pallas 

- 

1685.619 

Jupiter 

- 

4332.5S5117 

Saturn 

- 

-  10758.322161 

Uranus 

- 

30688.712687 

Mean  distances  from 

the 

Sun,  or  Semi-axes  of 

Orbits ,  the  Earth's  mean  distance  being  =  1. 

Mercury 

- 

0.387098 

Venus 

- 

-  0.723332 

The  Earth 

- 

1.000000 

Mars 

- 

-  1.523692 

Vesta  - 

- 

2.36319 

Juno 

- 

-  2.67035 

Ceres  - 

- 

2.76722 

Pallas 

- 

-  2.7718S 

Jupiter 

- 

5.202776 

Saturn 

- 

-  9.53S770 

Uranus 

- 

-  19.183305 

CHAPTER  XII, 


isp 


j Ratio  of  the  Eccentricity  to  the  Semi-transverse  axis, 
at  the  beginning  of  1801,  with  the  Secular  Varia¬ 
tion.  The  sign  —  indicates  a  diminution . 


Ratio  of  the 
Eccentricity. 

Secular  Variation! 

Mercury 

-  0.205515 

-  0.00000387 

Venus 

0.006861 

-  —  0.00006275 

The  Earth  - 

-  0.016853 

—  0.00004181 

Mars 

0.093307  - 

0.00009019 

Vesta  - 

-  0.089128 

Juno 

0.254311 

Ceres  - 

-  0.078502 

Pallas 

0.241600 

Jupiter 

-  0.048164 

0.00016036 

Saturn 

0.056132  - 

- 0.00031240 

Uranus 

-  0.046670 

—  0.00002521 

Mean  Longitudes,  reckoned  from  the  Mean  Equinox, 

at  the  Epoch  of  Mean  Noon,  at  Greenwich,  January 
1,  1801. 


Mercury 

166° 

0' 

49 

Venus 

-  11 

33 

3 

The  Earth 

100 

39 

10 

Mars 

-  64 

22 

56 

Vesta 

-  293 

32 

34 

Juno 

72 

55 

28 

Ceres  -  : 

77 

25 

23 

Pallas  - 

-  65 

22 

5 

Jupiter 

112 

15 

23 

Saturn 

-  135 

20 

18 

Uranus  r 

177 

47 

39 

188 


ASTRONOMY, 


Mean  Longitudes  of  the  Perihelia ,  for  the  same 
Epochs  as  the  preceding ,  with  the  Sidereal,  Secular 
Variations. 


Long.  Perihel. 

Sec 

:.  Var. 

Mercury 

74° 

21' 

47" 

-  9' 

44" 

Venus 

128 

43 

53 

.  —  4 

28 

The  Earth  - 

99 

30 

5  - 

-  19 

41 

Mars  -  *> 

332 

23 

57 

26 

22 

Vesta 

250 

18 

26 

Juno 

53 

13 

22 

Ceres 

146 

46 

32 

Pallas 

120 

54 

48 

Jupiter  ■» 

11 

8 

36 

11 

5 

Saturn 

89 

8 

58  - 

-  32 

17 

Uranus 

167 

21 

42 

4 

0 

Inclinations  of  the  Orbits  to  the  Ecliptic ,  at  the  be¬ 
ginning  of  1801,  with  the  Secular  Variations  of  the 
Inclinations  to  the  true  Ecliptic . 


Inclination. 

Sec.  Var. 

Mercury 

7° 

0' 

1" 

18".l 

Venus  - 

-  3 

23 

29  - 

—  4.5 

The  Earth 

0 

0 

0  - 

0 

Mars 

-  1 

51 

6 

—  0.3 

Vesta 

7 

7 

50 

Juno  - 

-  13 

4 

16 

Ceres 

10 

37 

31 

Pallas  - 

-  34 

35 

14 

Jupiter 

1 

18 

52 

—  22.6 

Saturn 

-  2 

29 

38  - 

-  —15.5 

Uranus 

0 

46 

26 

-  3.1 

CHAPTER  XII, 


189 


Longitudes  of  the  Ascending  Nodes  at  the  beginning 
of  1801j  with  the  Sidereal ,  Secular  Motions. 


Long,  of 

Sec.  Mot. 

Mercury  - 

45°  57'  31" 

—  13'  2" 

Venus  - 

74  52  39  - 

- 31  11 

The  Earth 

0  0  0 

-  0  0 

Mars  - 

48  0  3  - 

-  —  38  49 

Vesta 

103  9  51 

Juno 

171  8  29 

Ceres 

78  53  31 

Pallas  - 

172  32  31 

Jupiter  - 

98  26  18 

—  26  21 

Saturn  - 

111  55  46  - 

-  32  22 

Uranus  - 

72  51  14 

—  59  59 

Sidereal  Revolutions  of  the  Satellites , 

,  and  their  Mean 

Distances  from 

the  Planets  about  which  they  re - 

valve.  The  distances  are  expressed  in  terms  of  the 

Equatorial  Radius  of  the  Planet. 

JUPITER. 

- 

Mean  Dist. 

Sider.  Revol. 

1st  Satellite 

6.04853 

1.7691378  Days. 

2d  - 

-  9.62347  - 

3.5511810 

3d 

15.35024 

7.154552S 

4th  - 

-  26.99835  - 

16.6887697 

SATURN. 

Mean  Dist. 

Sider.  Revol. 

1st  Satellite 

3.351 

0.94271  Days 

2d  - 

-  4.300  - 

1 .37024 

3d  - 

5.284  - 

1.88780 

4th 

-  6.819  - 

-  2.73948 

5th 

9.524 

4.51749 

6th  - 

-  22.081  - 

-  15.94530 

7th 

64.359 

79.32960 

490 


ASTRONOMY. 

URANUS. 

Mean  Dist. 

Sider.  Revol* 

1st  Satellite 

13.120 

5.8926  Da ys 

2d  - 

-  17.022  - 

8.7068 

3d 

19.845 

10.9611 

4th  - 

-  22.752  - 

13.4559 

5th 

45.507 

38.0750 

6th  - 

-  91.008  - 

107.6944 

CHAPTER  XIII. 

On  Comets. 

1.  It  lias  already  been  said  (1.9)  that  a  Comet  is  a 
body  which  occasionally  appears  in  the  heavens,  has  a 
motion  among  the  fixed  stars,  and  only  continues  visi¬ 
ble  for  a  short  period.  The  appearance  of  a  comet  is 
usually  that  of  a  collection  of  vapour,  in  the  centre  of 
which  is  a  nucleus,  that  is,  in  general,  not  very  dis¬ 
tinctly  defined. 

The  motions  of  some  comets  are  direct,  and  of  others 
retrograde.  In  the  same  comet  the  motion  continues 
nearly  in  one  plane,  passing  through  the  sun’s  centre; 
but  for  different  comets  the  planes  make  very  different 
angles  with  the  ecliptic.  It  is  found  that  when  a 
comet  first  becomes  visible,  its  distance  from  the  sun  is 
is  diminishing;  and  that  when  it  ceases  to  be  visible, 
the  distance  is  increasing. 

3.  When  a  comet  first  appears,  its  nucleus  is  usually 
surrounded  by  a  faintly  luminous  vapour,  to  which  the 
name  of  Coma  has  been  given.  As  the  comet  ap¬ 
proaches  the  sun,  the  coma  becomes  more  bright,  and 
at  length  shoots  out  into  along  train  of  luminous  trans- 


CHAPTER  XIII.  191 

parent  vapour,  in  a  direction  opposite  to  the  sun.  This 
forms  the  Tail  of  the  comet. 

As  the  comet  recedes  from  the  sun,  the  tail  precedes 
it,  being  still  in  a  direction  opposite  to  the  sun,  and 
grows  less,  till  at  length  the  comet  resumes  nearly  its 
first  appearance.  In  those  comets  which  do  not  ap¬ 
proach  very  near  to  the  sun,  the  coma  does  not  extend 
into  a  tail.  The  tail  is  always  transparent,  so  that  the 
stars  are  distinctly  seen  through  it. 

4.  The  length  and  form  of  the  tail  are  very  various. 
In  some,  the  length  is  only  a  few  degrees,  and  in 
others  it  is  more  than  a  quadrant.  In  the  great  comet 
which  appeared  in  1680,  the  tail  extended  to  a  dis¬ 
tance  of  70°;  and  in  that  of  1618,  to  the  distance  of 
104°. 

5.  It  is  supposed  that  in  the  near  approach  of  a 
comet  to  the  sun,  the  heat  becomes  so  intense  as  to 
melt  and  evaporate  the  exterior  part,  and  thus  to 
form  round  the  interior,  an  atmosphere  of  vapour 
which  is  the  coma.  And  that  the  more  volatile 
parts  of  this  vapour  being  acted  on  by  the  impulsion  of 
the  sun’s  rays,  are  moved  in  a  direction  opposite  to  the 
sun,  and  thus  form  the  tail. 

6.  Comets  have  been  sometimes  observed  to  pass 
very  near  to  some  of  the  heavenly  bodies,  without  pro¬ 
ducing  any  sensible  effect  on  their  motions.  It  is  hence 
inferred  that  the  quantities  of  matter  which  they  con¬ 
tain  is  very  small. 

7.  A  comet  remains  so  short  a  time  in  sight,  and  de¬ 
scribes  so  small  a  part  of  its  course  within  our  view, 
that,  from  observation  alone,  without  the  assistance  of 
hypothesis,  it  would  be  impossible  to  determine  the  na¬ 
ture  of  its  path.  The  hypothesis  most  conformable  to 
analogy  is,  that  the  comet  moves  in  an  ellipse,  having 


192 


ASTRONOMY. 


the  sun  in  one  of  the  foci,  and  that  the  radius  vector 
describes  areas,  proportional  to  the  times. 

As  the  ellipse,  in  which  a  comet  moves,  is  evidently 
very  eccentric,  the  part  of  it  in  the  vicinity  of  the  ver¬ 
tex  or  perihelion,  and  through  which  the  comet  passes 
while  it  continues  visible,  must  coincide  very  nearly 
with  a  parabola. 

8.  The  elements  of  a  comet’s  orbit  are,  the  incli¬ 
nation  of  the  orbit,  the  position  of  the  line  of  the  nodes, 
the  longitude  of  the  perihelion,  the  perihelion  distance 
from  the  sun,  and  the  time  when  the  comet  is  in  the 
perihelion.  These  are  less  in  number  than  those  of  the 
orbit  of  a  planet  (12.31),  because  the  observations  that 
can  be  made  during  one  appearance  of  a  comet  are 
not  sufficient  to  determine  with  any  degree  of  accuracy, 
the  transverse  axis  of  the  orbit  and  the  periodic  time. 

9.  Assuming  the  orbit  of  a  comet  to  be  an  ellipse  or 
porabola  and  that  the  radius  vector  describes  areas 
proportional  to  the  times,  the  elements  may  be  deter¬ 
mined  from  three  observed,  geocentric  places  of  the 
comet.  This,  though  a  problem  of  considerable  dif¬ 
ficulty,  may  be  performed  in  a  great  variety  of  ways: 
almost  every  noted  astronomer  of  latter  time,  having 
given  a  method  of  his  own.  One  of  the  latest  and  in 
practice  one  of  the  best,  is  that  given  by  Delambre  in 
his  Jlstronomie ,  Chajp.  33,  art.  59,  &c. 

10.  The  only  comet  which  is  known  with  certainty 
to  have  returned,  is  that  of  1682,  which  conformably  to 
the  prediction  of  Dr.  Halley,  appeared  in  1759.  Halley 
was  led  to  this  prediction  by  observing  that  a  comet 
had  appeared  in  1531  and  another  in  1607*  and  that 
the  elements  of  their  orbits,  when  calculated  from  the 
observations  made  on  them,  agreed  nearly  with  those 
of  the  comet  of  1682.  A*e  thence  inferred  that  instead 


CHAPTER  XIV. 


193 


of  three  different  comets,  it  was  the  same  comet  that 
had  appeared  at  those  times,  and  that  its  period  was 
between  75  and  76  years. 

11.  The  number  of  comets  is  not  known,  but  it 
amounts  to  several  hundred. 

CHAPTER  XIV. 

•  I 

Aberration  of  Light,  Nutation  of  the  Earth’s  Axis , 

and  the  Annual  Parallax  of  the  fixed  Stars. 

\ 

1.  Dr.  Bradley  in  the  course  of  some  accurate  ob¬ 
servations  on  the  fixed  stars,  found  that  their  apparent 
places  were  subject  to  small  changes,  amounting  when 
greatest  to  about  40".  He  also  ascertained  that  those 
changes  were  annual,  as  their  magnitudes  were  the 
same  at  the  same  time  in  each  year.  These  observa¬ 
tions  were  commenced  in  the  year  1725,  and  continued 
for  several  years. 

After  several  unsuccessful  attempts,  to  explain  the 
cause  of  these  periodical  changes,  it  occurred  to  him 
that  in  consequence  of  the  progressive  motion  of  light, 
and  of  the  earth's  motion  in  its  orbit,  the  apparent  place 
of  a  star,  ought  generally  to  be  different  from  the  true 
place. 

2.  Let  OB,  Fig.  45,  be  a  portion  of  the  earth’s  or¬ 
bit,  so  small  that  it  may  be  considered  as  a  right  line, 
and  the  earth’s  motion  in  it,  uniform;  and  let  ES  be 
the  direction  of  a  fixed  star,  from  the  point  E.  Also 
let  AE  be  the  distance  through  which  the  earth  moves 
in  some  short  portion  of  time,  and  aE  the  distance 
through  which  a  particle  of  light  moves  in  the  same 
time.  Then  a  particle  of  light,  which,  coming  from 
the  star  in  the  direction  SE,  is  at  a ,  at  the  same  time 

36 


194 


ASTRONOMY. 


that  the  earth  is  at  A,  will  arrive  at  E,  at  the  same 
time  that  the  earth  does.  Let  A  A  "a"  and  ES'  be 
each  parallel  to  A  a.  Then  ao!  is  to  ?AA'  and  a' a"  is 
to  A' A"  in  the  ratio  of  Ea  to  EA.  Consequently 
when  the  earth  is  at  A',  the  particle  of  light  is  at  a', 
and  when  the  earth  is  at  A",  the  particle  of  light  is  at 
a".  The  particle  of  light  therefore,  continues  in  the 
same  direction  from  the  earth,  that  is,  in  the  direction 
A  a  or  ES.'  Hence  it  meets  the  earth  at  E,  in  the  di¬ 
rection  S'E.  To  an  eye  at  E,  the  particle  of  light  en¬ 
tering  it  in  the  direction  S'E,  appears  to  come  from  a 
star  in  the  direction  ES'.  What  has  been  said  of  a 
single  particle,  will  apply  to  all  the  particles  coming 
from  the  star,  and  entering  the  eye.  Consequently  the 
star  appears  to  be  in  the  direction  ES'. 

3.  The  angle  which  expresses  the  change  pro¬ 
duced  in  the  apparent  place  of  a  body,  by  the  mo¬ 
tion  of  light  combined  with  the  motion  of  the  spectator, 
is  called  the  Aberration .  Thus  S'ES  is  the  aberration 
of  the  star  S. 

4.  Various  formulae  have  been  investigated  for  com¬ 
puting  the  effect  of  aberration  on  the  longitudes,  lati¬ 
tudes,  right  ascensions  and  declinations  of  the  heavenly 
bodies,  and  particularly  of  the  fixed  stars.  Of  those 
that  apply  to  the  fixed  stars,  the  following  are  some  of 
the  most  simple. 

5.  Let  L  be  the  longitude  of  the  sun  at  the  time  for 
which  the  aberration  is  required,  and  L'  and  A,  the 
longitude  and  latitude  of  the  star.  Then, 

,,  .  r  —20".  253  cos  (L  — L') 

Jibtr.  m  Long.  = - ^ - 

cos  a 

Aber.  in  Lat.  —  20",253  sin  (L'  —  L)  sin  a. 

6.  Let  A  be  the  right  ascension  and  D  the  declina- 


CHAPTER  XIV.  195 

tion  of  the  star,  L  being  the  sun’s  longitude  as  before. 
Then, 

Aber.  in  Right  Jlscen. 

_  0".837  cos  (A  +  L)  —  19".416  cos  (A  —  L) 

cos.  D. 

Aber.  in  Decl.  —  sin  D  [19". 416  sin  (A —  L)  —  0".837  sin 
(A  4-  L)  —  8".0G6  cos  L  cos  D] 

7.  In  catalogues  of  the  fixed  stars,  the  mean  places 
are  given.  By  means  of  the  preceding  formula,  or  by 
small  tables  which  have  been  calculated  for  the  pur¬ 
pose,  the  aberration  may  be  found,  and  thence  the  ap¬ 
parent  place  of  the  star. 

8.  In  consequence  of  the  aberration,  each  star  ap¬ 
pears  to  describe  an  ellipse  in  the  heavens,  of  which 
the  true  place  is  the  centre;  the  semi- transverse  axis  is 
&0".253  and  the  semi-congugate  is  20 ".253  sin  a. 

9.  The  supposition  of  the  earth’s  annual  motion, 
serves  fully  to  explain  the  phenomena  of  the  aberra¬ 
tion.  And  the  amounts  of  the  aberrations  for  different 
stars,  and  at  different  times,  computed  on  that  suppo¬ 
sition,  are  found,  exactly  to  agree  with  observation. 
These  circumstances  form  the  strongest  proof  of  the 
reality  of  the  earth’s  annual  motion. 

10.  The  aberration  of  the  sun,  which  has  place 
only  in  longitude  is  —  20".253.  Thus  the  sun’s  ap¬ 
parent  place  is  always  about  20 ".25,  behind  its  true 
place.  Solar  tables  give  the  apparent  place  of  the 
sun,  as  affected  by  aberration,  and  it  is  this  which  is 
generally  wanted. 

11.  For  a  planet,  the  aberration  is  different  from 
what  it  is  for  a  fixed  star;  because  the  planet  changes 
its  place  during  the  time  that  light  is  passing  from  it 


196 


ASTRONOMY. 


to  the  earth.  The  aberration  is  therefore  increased  or 
diminished  by  the  geocentric  motion  of  the  planet 
during  this  time. 

For  the  moon,  the  aberration  is  always  very  small, 
only  amounting  to  a  fraction  of  a  second. 

12.  Besides  the  aberration  produced  by  the  annual 
motion  of  the  earth,  there  is  another,  called  the  Diurnal 
aberration,  which  is  produced  by  the  earth’s  motion 
on  its  axis.  This  is  however  so  small  as  to  be  nearly 
insensible. 

NUTATION. 

13.  Small  inequalities,  w  hich  have  been  observed 
in  the  precession  of  the  equinoxes,  and  in  the  mean 
obliquity  of  the  ecliptic;  are  called  Nutation.  These 
inequalities  were  discovered  by  Dr.  Bradley  while 
employed  in  verifying  his  theory  of  the  aberration. 

14.  The  period  of  the  changes  of  these  inequalities 
w  as  observed  to  be  about  the  same  as  the  period  of  the 
revolution  of  the  moon’s  nodes;  and  it  was  found  that 
the  quantities  of  the  inequalities  depended  on  the  place 
of  the  node. 

15.  The  phenomena  of  the  nutation  may  be  repre¬ 
sented  by  supposing,  that  while  a  point,  which  may 
be  considered  as  defining  the  mean  place  of  the  pole 
of  the  equator,  describes  a  circle  in  the  heavens,  round 
the  pole  of  the  ecliptic,  at  a  distance  from  it,  equal  to 
the  mean  obliquity  of  the  ecliptic,  and  with  a  retro¬ 
grade  motion  of  50".  1  annually,  another  point  repre¬ 
senting  the  true  pole  of  the  equator,  moves  round  the 
former  at  the  distance  of  9",  so  as  to  be  always  90° 
more  easterly  than  the  moon’s  ascending  node.  The 
inequalities  thus  produced  in  the  precession  of  the 
equinoxes,  and  in  the  obliquity  of  the  ecliptic,  will 


CHAPTER  XIV.  197 

very  nearly  agree  with  the  observed  inequalities.  The 
agreement  becomes  more  exact,  if  instead  of  supposing 
the  true  pole  to  describe  a  circle  about  its  mean  place, 
it  be  supposed  to  describe  an  ellipse,  having  its  semi- 
transverse  axis  equal  9". 6  and  its  semi-conjugate  7". 5. 

16.  If  N  be  the  longitude  of  the  moou’s  ascending 
node,  the  variation  in  the  obliquity  of  the  ecliptic,  pro¬ 
duced  by  the  mutation  is  -f  9". 6  cos  N;  the  inequality 
in  the  motion  of  the  equinoxes  in  longitude,  sometimes 
called  the  Equation  of  the  Equinoxes  in  Longitude ,  is 
17"-9I6  sin  N;  and  the  inequality  in  tlieir  motion  in 
right  ascension,  called  the  Equation  of  the  Equinoxes 
in  Eight  Ascension ,  is  —  16".462  sin  N. 

17.  The  equation  of  the  equinoxes  in  longitude, 
equally  effects  the  longitudes  of  all  the  stars.  The 
equation  of  the  equinoxes  in  right  ascension  also  affects 
the  right  ascensions  of  all  the  stars,  but  it  only  forms  a 
part  of  the  nutation  in  right  ascension. 

18.  If  A  be  the  right  ascension  of  a  star,  and  D  its 
declination,  then, 

Nutation  in  Right  Ascen.  — 

—  16".462  sin  N  —  8".373  cos  (A  —  N)  tan  D 

—  1'227  cos  (A  4-  N)  tan  D. 

Nutation  in  Decl.  = 

+  8  '.373  sin  (A  —  N)  +  l". 227  sin  (A  +  N), 

19.  The  nutation  does  not  affect  the  positions  of  the 
stars  relative  to  oue  another,  nor  to  the  plane  or  pole  of 
the  ecliptic;  it  only  affects  tlieir  positions  relative  to  the 
plane  of  the  equator,  or  to  the  position  of  the  earth’s 
axis. 

ANNUAL  PARALLAX  OF  THE  FIXED  STARS. 

20.  The  angle  contained  between  two  straight  lines, 
conceived  to  be  drawn  from  the  sun  and  earth,  and 


19$ 


ASTRONOMY. 


meeting  at  a  fixed  star  is  called  the  Annual  Parallax 
of  the  star. 

SI.  Observations  have  been  made  by  several  astro¬ 
nomers,  on  different  stars,  and  at  times  when  the 
earth  was  in  opposite  parts  of  its  orbit,  with  the 
view  of  ascertaining  whether  they  have  any  sensible 
parallax.  Dr.  Brincly  deduced  from  his  observations, 
that  the  parallax  of  Lyrae ,  when  greatest,  that  is, 
when  the  line  joining  the  sun  and  earth  is  perpendicu¬ 
lar  to  the  line  joining  the  sun  and  star,  is  nearly  3". 
But  Pond,  the  present  Astronomer  Royal  of  England, 
from  a  series  of  very  accurate  observations  on  the 
same  star,  makes  the  greatest  parallax  only  0".26. 

22.  If  we  suppose  the  annual  parallax  of  a  star, 
when  greatest  to  be  1",  the  distance  of  the  star  will 
be  206265  times  the  radius  of  the  earth’s  orbit.  This 
distance  is  so  immensely  great,  that  light,  which  tra¬ 
verses  the  distance  from  the  sun  to  the  earth  in  8  m. 
13  sec.  would  require  more  than  3  years  to  come  from 
the  star  to  the  earth. 


CHAPTER  XV. 

Nautical  Astronomy, 

1.  Some  of  the  most  useful  practical  applications  of 
astronomy  are  those  which  serve  to  make  known  to  the 
Navigator,  his  latitude  and  longitude,  when  at  sea. 
The  continual  agitation  in  the  motion  of  a  ship  does 
not  permit  the  use  of  instruments,  which  are  adjusted 
by  a  plumb  line  or  spirit  level.  The  astronomical  in¬ 
struments  used  at  sea,  are  the  Hadley's  Quadrant ,  the 
Sextant  and  Circle  of  Reflection,  By  either  of  these 
the  altitude  of  any  of  the  heavenly  bodies,  and  the  an- 


CHAPTER  XV. 


199 


gulUr  distance  between  them,  within  certain  limits,  may 
be  obtained  with  considerable  accuracy.  The  sextant 
and  circle  of  reflection  are  made  with  greater  accuracy 
than  the  quadrant,  and  are  principally  used  for  mea¬ 
suring  the  angular  distance  of  the  moon  from  the  sun 
or  a  fixed  star.  For  descriptions  of  these  instruments 
and  of  the  methods  of  adjusting  and  using  them,  the 
student  is  referred  to  Bowditch’s  Practical  Navigator. 

2.  The  Mile  used  in  measuring  distances  at  sea,  is 
the  60th  part  of  a  degree.  So  that  a  mile  just  corres¬ 
ponds  to  a  minute. 

3.  The  course  on  which  a  ship  sails  is  determined 
by  an  instrument  called  a  Mariner’s  Compass;  and 
the  rate  at  which  she  sails  by  an  instrument  called  a 
Log .  The  latter  is  a  piece  of  board  in  the  form  of  a 
sector  of  a  circle,  the  circular  part  of  which  is  loaded 
with  lead,  so  that  when  in  the  water  it  may  keep  a 
verticle  position.  To  the  log  is  attached  a  line  of  con¬ 
siderable  length,  divided  into  spaces  called  Knots ,  and 
and  wrapped  round  a  light  reel.  The  length  of  a  knot 
is  such,  that  when  the  log  is  thrown  into  the  sea,  and 
the  line  allowed  to  run  freely  off  the  real,  the  number 
of  knots  which  pass  off  in  a  half  minute,  indicates 
the  number  of  miles  the  ship  is  then  sailing  in  an 
hour. 

4.  That  portion  of  the  surface  of  the  ocean,  which 
a  ship  traverses  during  a  few  hours,  or  even  during  a 
day,  does  not  differ  much  from  a  plane.  Supposing 
it  to  be  a  plane,  and  also  that  the  meridians  are  paral¬ 
lel  to  one  another,  let  AB,  j Fig.  46,  be  the  track  of  the 
ship,  NS  and  N'S',  meridians  passing  through  A  and 
B,  and  AO  and  BD  parallels  of  latitude,  Then  in 
the  triangle  DAB,  we  have,  ) 


200 


ASTRONOMY. 


AD  —  AB  cos  DAB, 

Or,  Diff.  of  Lat.  =  Dist.  x  cos.  Course. 

It  is  proved  by  writers  on  the  theory  of  navigation, 
that  this  expression  for  the  difference  of  latitude  is  ri¬ 
gidly  true,  even  when  the  earth  is  considered  as  a  sphere 
or  spheroid.  But  in  consequence  of  currents  and 
other  causes,  the  distance  and  course  can  never  he  ob¬ 
tained  with  great  accuracy,  and  consequently  the  dif¬ 
ference  of  latitude  thus  found,  must  he  considered  only 
as  an  approximation. 

5.  The  triangle  ADB,  also  gives, 

BD  =  Dist.  x  sin.  Course. 

As  the  meridians  are  not  parallel,  but  really  con¬ 
verge  towards  the  poles,  each  way  from  the  equator,  it 
is  evident  that  BD  is  greater  than  the  distance  be¬ 
tween  the  meridians  on  one  of  the  parallals  of  lati¬ 
tude  and  less  than  that  on  the  other,  except  when  A 
and  B  are  on  opposite  sides  of  the  equator  and  equally 
distant  from  it.  It  is  in  general  nearly  equal  to  the 
distance  between  the  meridians  on  a  parallel  of  lati¬ 
tude,  midway  between  the  parallels  passing  through 
A  and  B.  The  latitude  of  this  parallel  is  called  the 
Middle  Latitude ,  and  is  equal  to  half  the  sura  or  half 
the  difference  of  the  latitudes  of  A  and  B,  according 
as  they  are  on  the  same,  or  on  opposite  sides  of  the 
equator. 

6.  If  AB,  Fig.  36,  be  considered  as  a  part  of  the 
equator,  P  its  pole,  ED  a  parallel  of  latitude,  and 
PDA,  PEB  meridians,  passing  through  any  two 
places,  then  (11.85)  AB.  the  difference  of  longitude  of 
the  places  is  equal  to  ED  divided  by  the  cosine  of  BE* 
Hence  (5),  Fig.  46, 


CHAPTER  XV. 


SOI 


Biff.  of  Long,  of  A  and  B 

BD  _  Dist.  x  sin.  Course 
cos  Mid  Lat.  cos  Mid.  Lat. 

7.  The  computed  differences  of  longitude  and  lati¬ 
tude,  applied  to  the  longitude  and  latitude  of  the  place 
A,  give  nearly  those  of  the  place  B,  when  the  distance 
between  the  places  is  not  great.  The  longitude  and 
latitude  thus  found  are  called  the  Estimated  longitude 
and  latitude. 

8.  As  the  longitude  and  latitude,  found  in  the  pre¬ 
ceding  manner  can  not  he  depended  on,  except  for  a 
short  time,  it  is  necessary  that  the  navigator  should  be 
able  to  determine  them  by  observation.  When  the 
weather  is  favourable  the  latitude  is  determined  each 
day  at  noon,  by  observation  of  the  sun’s  altitude  about 
that  time.  Several  altitudes  being  observed,  it  is  easy 
to  infer  the  greatest  altitude  that  the  sun  acquires, 
which  is  the  meridian  altitude.  But  this  altitude  is  the 
apparent  altitude  of  the  under  or  upper  limb,  usually 
the  former,  and  must  be  corrected  for  refraction,  paral¬ 
lax  and  semidiameter.  It  also  requires  another  cor¬ 
rection.  The  observation  gives  the  altitude  above  the 
visible  horizon;  and  consequently,  as  the  observer  is 
on  the  deck  of  the  vessel,  several  feet  above  the  sur¬ 
face  of  the  water,  it  is  too  great.  A  small  table  ac¬ 
companies  every  treatise  on  navigation,  containing  a 
correction,  depending  on  the  height  of  the  eye,  which 
is  to  be  subtracted  from  the  observed  altitude.  This 
correction  is  called  the  Dip  of  the  Horizon. 

From  the  correct  meridian  altitude  of  the  sun,  the 
latitude  is  easily  determined  (42). 

9.  Sometimes  the  sun  is  hid  by  clouds,  so  as  to 
prevent  the  observation  of  the  meridian  altitude,  and 
yet  it  is  visible  at  other  times  in  the  day.  In  such 

27 


203 


ASTRONOMY. 


cases  the  latitude  may  he  found  from  two  observed  al¬ 
titudes  with  the  interval  of  time  between  them.  The 
interval  ought  if  possible  to  be  two  or  three  hours;  and 
one  of  the  altitudes  should  be  as  near  to  noon  as  cir¬ 
cumstances  will  admit. 

10.  When  the  altitudes  are  taken  at  different  places, 
as  is  generally  the  case  at  sea.  the  less  one  should  be 
reduced  to  what  it  would  have  been,  if  it  had  been 
taken  at  the  same  instant,  at  the  place  where  the  other 
is  taken. 

This  may  be  clone  with  sufficient  accuracy  in  a  very  simple 
manner.  Let  A,  Fig.  47,  be  the  place  where  the  less  altitude  is 
taken,  B  that  where  the  greater  is  taken,  and  AS  the  line  in  which 
the  horizon  is  intersected  by  a  vertical  circle  passing  through  A 
and  the  sun,  at  the  time  of  observing  the  less  altitude.  From 
the  bearing  of  the  sun  at  that  time,  and  the  course  the  ship  is 
sailing,  the  angle  BAS  is  known.  On  AS  let  fall  the  perpendicu¬ 
lar  BD.  Then  it  is  evident  that  the  altitudes  of  the  sun,  at  B  and 
D,  at  the  same  time,  are  the  same.  But  the  altitude  at  D  is 
greater  than  the  altitude  at  the  same  time  at  A,  by  the  number  of 
minutes  contained  in  AD.  Consequently  AD  is  the  correction 
of  the  less  altitude.  Hence  the 

Correction  =  AD  =  AB  cos  BAS. 

When  the  angle  BAS  is  greater  than  90°,  the  cosine  is  negative, 
and  the  correction  must  be  subtracted  from  the  altitude. 

11.  Given  two  altitudes  of  the  sun ,  with  the  interval  of  time  be - 
Ween  the  observations ,  to  find  the  latitude  of  the  place. 

Let  Z,  Fig  48,  be  the  zenith  of  the  place  at  which  the  greater 
altitude  is  taken,  P  the  pole,  S  the  place  of  the  sun  at  the  time  of 
the  less  altitude,  and  S'  its  place  at  the  time  of  the  greater.  As 
the  sun’s  declination  changes  but  little  in  the  course  of  a  few 
hours,  PS  and  PS'  may  be  each  considered  as  equal  to  the  sun’s 
polar  distance  at  the  middle  of  the  time  between  the  observations; 
and  consequently  the  triangle  PSS'  may  be  regarded  as  isosceles. 
If  PG  be  perpendicular  to  SS',  it  will  bisect  it  in  G.  Put, 


CHAPTER  XV. 


203 


H  =  ES  =  the  less  altitudes,  reduced, 

H'  =  FS'  =  the  greater  altitude, 

D  =  PS  =  90°  ±  sun’s  declination, 

A  =  SPS'  =  interval  of  time,  expressed  in  degrees, 

L  =  HP  =  latitude,  where  the  greater  altitude  is  taken, 
U  =  PS Z,  V  =  ZSS',  W  =  PSS',  and  X  =  SS'. 


Then  from  the  right  angled  triangle  PSG,  we  have, 

sin  i  X  ==  sin  h  A  sin  D,  and  cot.  W  =  cos  D  tan  §  A. 

From  the  triangle  ZSS',  by  an  investigation  exactly  similar  to 
that  in  article  20th,  chap.  9th.  we  have, 

/H  +  X  +  H'\  .  /H  +  X  +  H'  „\ 

sin  i  V  =  v/  _ _ '  1  2 _ 

cos  H  sin  X 
Then  U  =  Wq:V 

The  upper  sign  has  place,  when  the  sun  passes  the  meridian 
on  the  opposite  side  of  the  zenith  from  the  elevated  pole,  and  the 
under,  when  it  passes  on  the  same  side. 


Now  from  the  triangle  PSZ, 

cos  ZP  =  cos  PS  cos  ZS  +  cos  PSZ  sin  PS  sin  ZS, 

Or,  sin  L  =  cos  D  sin  H  -f  cos  U  sin  D  cos  H 

=  cos  D  sin  H  -f-  (2  cos2  5  U  —  I  sin  D  cos  H  (App.  9), 
=  2  cos2  h  U  sin  D  cos  H  —  (sin  D  cos  H  —  cos  D  sin  H), 
=  2  cos2  h  U  sin  D  cos  H  —  sin  (D  —  H). 

But, 

sin  L  =  —  cos  (90°  +  L)  =  1  —  2  cos2  \  (90  4-  L)  (App.  9) 
and,  sin  (D  —  H)  =  cos  [90°  _  (D  —  H)]  =  2  cos2  h  [90°  — 
(D  — H)]  — 1. 

By  substituting  these  values  we  have, 
cos2  \  (90°  4-  L)  =  cos2  5  [90°  —  (D  —  H)]  —  cos2  5  U  sin  D 
cos  H 


=  cos2  i  [90°  —  (D  —  H)].  (1 


cos2  5  U  sin  D  cos  H  ^ 
cos2  h  [90°  —  (D  —  H)]  / 


Make  sin  M  = 


cos  \  U  sin  D  cos  H 
cosi  [90°  —  (D  —  H)] 


204 


ASTRONOMY. 


Then  cos2  \  (90°  +  L)  cos2  \  [90  —  (D  —  H  ].  (1  —  sin2  M) 
=*  cos2  \  [90°  —  D  —  H)]  cos2  M. 

Or,  cos  i  (90°  +  Lj  =  cos  \  [90°  —  (D  —  IT]  cos.  M. 

12.  When  the  latitude  is  determined,  the  time  may 
be  obtained  by  an  observation  of  the  sun’s  altitude,  a 
few  hours  from  noon  (9. 20).  Supposing  the  watch  or 
chronometer,  used  on  board  the  vessel  to  have  been 
•well  regulated  and  set,  previously  to  leaving  port,  and 
that  it  keeps  time  accurately,  the  difference  between  the 
time  obtained  from  observation,  and  that  shown  by  the 
watch,  gives  the  difference  of  longitude,  in  time.  But 
the  best  time  keeper  can  not  be  entirely  depended  on, 
and  therefore  the  longitude,  thus  obtained,  is  liable  to 
u  ncertainty. 

13.  If  the  true  angular  distance  between  the  centres 
of  the  moon  and  sun,  or  between  the  centre  of  the 
moon  and  some  star,  near  the  ecliptic,  be  obtained 
from  calculations,  founded  on  the  observed  angular 
distance;  and  the  time  when  they  are  at  that  distance, 
be  determined  by  calculation  for  the  meridian  of  Green- 
wich;  then  the  difference  between  the  calculated  time, 
and  the  time  of  observation  as  reckoned  at  the  meri¬ 
dian  of  the  ship,  will  give  the  longitude  from  Green¬ 
wich. 

14.  The  Nautical  Almanac  contains  the  distances  of 
the  moon  from  the  sun,  and  from  several  stars  that  are 
best  adapted  to  the  purpose.  The  distances  are  given 
to  every  third  hour.  It  is  therefore  easy  to  determine 
by  proportion,  the  time  when  either  distance  is  of  a 
given  magnitude. 

15.  The  observed  distance  must  be  corrected  for  the 
semidiameter  of  the  moon  when  the  observation  is  of 
the  moon  and  a  star,  and  for  the  semidiameter  of  the 
sun  and  moon,  when  it  is  of  those  bodies,  so  as  to  give 


CHAPTER  XV. 


205 


the  apparent  distance  of  the  centres.  To  obtain  the 
true  distance,  the  apparent  distance  must  be  corrected 
for  the  effects  of  refraction  and  parallax.  This  re¬ 
quires  that  the  altitudes  of  the  bodies  should  be 
known. 

16.  The  altitudes  may  be  taken  by  two  assistant  ob¬ 
servers,  at  the  same  time  that  the  principal  one  ob¬ 
serves  the  angular  distance.  If  there  is  but  one  ob¬ 
server,  he  can  first  take  several  altitudes  of  the  bodies; 
then  several  distances;  and  afterwards  several  more 
altitudes,  noting  the  times  of  all  the  observations. 
Thence  it  it  easy  to  infer,  with  sufficient  accuracy, 
the  altitudes  corresponding  to  the  mean  of  the  dis¬ 
tances. 

1 7.  Given  the  apparent  distance  of  the  moon  and  sun ,  or  of  the 
moon  and  a  fixed  star ,  and  the  altitudes  of  the  bodies ,  to  determine 
the  true  distance. 

Let  Z,  Fig.  49,  be  the  zenith,  ZH  the  vertical  passing  through 
the  moon,  and  ZO,  that  passing  through  the  sun,  or  a  star.  Then 
as  the  moon  is  more  depressed  by  parallax,  than  it  is  eleva¬ 
ted  by  refraction,  the  apparent  place  is  below  the  true  place.  But 
for  the  sun  or  a  star  as  the  parallax  is  very  little  or  insensible,  the 
apparent  place  is  above  the  true  place.  Let  M  be  the  apparent 
place  and  M'  the  true  place  of  the  moon;  and  S  the  apparent 
place  and  S'  the  true  place  of  the  sun  or  star.  Put, 

IJ  =  HM  =  apparent  altitude  of  the  moon, 

H'  =  HM'  =  true  do. 

A  =  OS  =  apparent  altitude  of  the  sun  or  star, 

A'  =  OS'  =  true  do. 

D  =  MS  =  apparent  distance, 

D'  =  M'S'  =  true  do. 

Then  in  the  triangle  ZMS,  we  have,  (App.  34), 


206 


ASTRONOMY. 


cos  Z  = 


cos  D  — sin  H  sin  A 


(App.  14) 


cos  H  cos  A 

__  cos  D  —  cos  H  cos  A  -f  cos  HI  4-  A) 
cos  H  cos  A 

__  cos  D  4-  cos  H  4-  A)  _ j 

cos  H  cos  A 

In  like  manner  from  the  triangle  ZM'S'  we  have, 

rr  cosD'  4-  COS  (H'  +  A')  . 

cos  Z* - - - — - —  1 . 

cos  hi  cos  A 

Hence  cos  ^  +  cos  (H  4-  A)  _  cos  D'  4-  cos  (H'  4-  A') 
cos  H  cos  A  cos  H'  cos  A' 

,  COS  H  COS  A  r  ipv  ,  /TT  I  A  \1 

Or,  cos  D  = - — - — .  [cos  D  4-  cos  (H  4-  A)]  — 

cos  H  cos  A 

cos  (H'  4-  A'). 

But, 

cos  D  4-  cos  (H  4-  A)  =  2  cos  \  (H  4-  A  4  D)  cos  \  (H  4- 
A  —  D),  (App.  22) 

cos  (IT  4  A')  =  2  cos2  \  (H'  4-  A')  —  1  (App.  9) 
cos  D  ==  1  —  2  sin2  \  D'  (App.  8). 

Substituting  these  values^  and  reducing,  we  have, 
sin2  h  D'  ==  cos2  |fH'  4  A')  — 

cos  \  (H  A  4-  D)  cos  \  (H  4-  A  —  D)  cos  H'  cos  A' 
cos  H  cos  A 

=e  cos2  \  (H'  4-  A'). 

cos  5  (H  4  A  4-  D)  cos  \  (H  4-  A  —  D)  cos  H'  cos  A' 


cos  H  cos  A 


cos2  h  (H'  -f  A') 

Make  sin  M  = 

cos  5  (H  4-  A  4-  D)  cos -2  (H  4-  A  —  D)  cos  H'  cos  A.^ 


n/ 


cos  H  cos  A 


cos  I  (H'  4-  A') 

Then, 

sin2  \  D'  =  cos2  (H  4-  A').  (1  —  sin2  M;  =  cos2  \  (H'  + 
A')  cos2  M, 

Or,  sin  \  D  =  cos  (H'  4-  A')  cos  M. 


CHAPTER  XVI. 


SO? 


CHAPTER  XVI. 

Of  the  Calendar . 

1.  The  Calendar  is  a  distribution  of  time  into 
periods  of  different  lengths,  as  years,  months,  weeks, 
and  days. 

S.  It  has  been  shown  that  the  tropical  year  contains 
36 5  d.  5  h  48  ni.  51.6  sec.  (7.9).  But  in  reckoning 
time  for  the  common  purposes  of  life,  it  is  most  con¬ 
venient  to  have  the  year  to  contain  a  certain  number  of 
whole  days.  In  the  calendar  established  by  Julius 
Caesar,  and  thence  called  the  Julian  calendar,  three 
successive  years  are  made  to  consist  of  365  days,  each; 
and  the  fourth,  of  366  days.  The  year,  which  con¬ 
tains  366  days,  is  called  a  Bissextile  year.  It  is  also 
frequently  called  Leaj)  year.  The  others  are  called 
Common  years.  The  added  day  in  a  bissextile  year 
is  called  the  Intercalary  day. 

3.  According  to  the  Julian  Calendar,  and  reckoning 
from  the  epoch  of  the  Christian  era,  every  year,  the 
number  of  which  is  exactly  divisible  by  4,  is  a  bissex¬ 
tile;  and  the  others  are  common  years. 

4.  It  is  evident  that  the  reckoning  by  the  Julian 
calendar,  supposes  the  length  of  the  year  to  be  365J 
days.  A  year  of  this  length  is  called  a  Julian  Year . 
A  Julian  year,  therefore,  exceeds  the  true  astrono¬ 
mical  year,  by  11  m.  8.4  sec.  This  difference  amounts 
to  rather  more  than  a  day,  in  130  years. 

5.  At  the  time  of  the  Council  of  Nice,  which  was  held 
in  the  year  325,  the  Vernal  Equinox  fell  on  the  2  ( st  of 
March,  according  to  the  Julian  calendar.  But  by  the  lat¬ 
ter  part  of  the  1 6th  century,  in  consequence  of  the  ex¬ 
cess  of  the  Julian  year  above  the  true  solar  year,  it  came 


208 


ASTRONOMY. 


ten  days  earlier*  that  is,  on  the  1 1th  of  March.  It  was 
observed  that  by  continuing  to  reckbti  according  to  the 
Julian  calendar,  the  seasons  would  fall  back,  so  that 
in  process  of  time  they  would  correspond  to  quite  dif¬ 
ferent  times  of  the  year.  This  reckoning  also  led  to 
irregularity  in  the  times  of  holding  certain  festivals  of 
the  church.  The  subject,  claiming  the  attention  of 
Pope  Gregory  XIII.  he,  with  the  assistance  of  several 
astronomers,  reformed  the  calendar.  To  allow  for 
the  10  days,  by  which  the  vernal  equinox  had  fallen 
back  from  the  2 1st  of  March,  he  ordered  that  the  day 
following  the  4th  of  October  1582.  should  be  reckoned 
the  15th,  instead  of  the  5th.  And  in  order  to  keep 
the  vernal  equinox  to  the  21  st  of  March,  in  future,  it 
was  concluded  that  three  intercalary  days  should  be 
omitted  every  four  hundred  years.  It  was  also  con¬ 
cluded  that  the  omission  of  the  intercalary  days  should 
take  place  in  those  centurial  years,  the  numbers  of 
which,  were  not  divisible  by  400.  Thus  the  years 
1700,  1800,  and  1900,  which,  according  to  the  Julian 
calendar  would  be  bissextiles,  would,  according  to  the 
reformed  calendar,  be  common  years. 

6.  The  calendar,  thus  reformed,  is  called  the  Grego¬ 
rian  Calendar .  It  is  easy  to  perceive,  by  a  short  cal¬ 
culation,  that  time  reckoned  by  the  Gregorian  calen¬ 
dar,  agrees  so  nearly  with  that  reckoned  by  true  solar 
years,  that  the  difference  does  not  amount  to  a  day  in 
4000  years. 

7.  The  Gregorian  calendar  was  at  once  adopted  in 
Catholic  countries,  but  in  those,  where  the  Protestant 
Religion  prevailed,  it  did  not  obtain  a  place,  till  some 
time  after.  In  England  and  her  colonies,  it  was  not 
introduced  till  the  year  t7^2.  It  is  now  used  in  all 
Christian  countries,  except  Russia. 


CHAPTER  XVJ. 


209 


8.  The  Julian  and  Gregorian  calendars  are  also  de¬ 
signated  by  the  terms  Old  Style  and  JSF ew  Style.  In 
consequence  of  the  intercalary  days,  omitted  in  the 
years  1700  and  1800,  there  is  now  12  days  difference 
between  them. 

9.  The  year  is  divided  into  twelve  portions,  called 
calendar  months.  Each  of  these  contain,  either  30  or 
31  days,  except  the  second  month,  February,  which 
in  a  common  year,  contains  28  days,  and  in  a  bissex¬ 
tile,  29  days;  the  intercalary  day  being  added  at  the 
last  of  this  month. 

10.  It  was  formerly  customary  to  designate  the  days 
of  the  week  in  the  calendar  by  the  first  seven  letters  of 
the  alphabet,  always  placing  them  so,  that  A  corres¬ 
ponded  to  the  first  day  of  the  year,  B  to  the  second,  C 
to  the  third,  D  to  the  fourth,  E  to  the  fifth,  F  to  the 
sixth,  G  to  the  seventh,  A  to  the  eighth,  B  to  the  ninth, 
and  so  on.  According  to  this  arrangement,  whatever 
letter  designates  any  given  day  of  the  week  in  the  first 
part  of  the  year,  continues  to  designate  the  same, 
throughout  the  year.  The  letter  designating  the  first 
day  of  the  week,  or  Sunday,  is  called  the  Dominical 
Letter . 

11.  As  a  common  year  consists  of  365  days,  or  52 
weeks  and  1  day,  the  last  day  of  each  year  must  fall 
on  the  same  day  of  the  week  as  the  first,  and  the  next 
year  must  commence  one  day  later  in  the  week.  Con¬ 
sequently  the  day  of  the  week  which  was  the  first  day 
of  the  former  year,  and  was  designated  by  A,  is  the 
seventh  day  of  the  second,  year,  and  is  designated  by 
G;  that  which  was  the  second,  and  was  designated  by 
B,  in  the  former  year,  is  the  first,  and  is  designated  by 
A  in  the  second,  and  so  ou.  It  therefore  follows,  that 
whatever  letter  is  the  dominical  letter,  in  any  common 


210 


ASTRONOMY. 


year,  the  letter  next  preceding  it  in  the  alphabet,  is  the 
dominical  letter  in  the  following  year,  except  the  for¬ 
mer  was  A,  in  which  case  the  second  is  G. 

12.  In  every  common  year,  the  first  day  of  March, 
is  the  60th  day  of  the  year,  and  consequently  corres¬ 
ponded  to  the  letter  D.  In  bissextile  years,  on  ac¬ 
count  of  the  intercalation,  the  1st  of  March  is  the  Gist 
day  of  the  year;  but  i he  letter  D  was  still  made  to 
correspond  to  it,  and  the  letters  for  the  remaining  part 
of  the  year  were  arranged  accordingly.  It  therefore 
follows  that,  after  the  29th  of  February,  any  given  day 
of  the  week  was  designated  by  the  letter  in  the  alpha¬ 
bet,  next  preceding  that,  by  which  it  was  designated 
in  the  first  two  months.  Consequently  a  bissextile 
had  two  dominical  letters,  one  of  which  appertained  to 
January  and  February,  and  the  other,  which  was  the 
next  preceding  letter  in  the  alphabet,  appertained  to 
the  other  ten  months. 

13.  From  w  hat  has  been  said,  it  follows  that  the  do¬ 
minical  letters  succeed  one  another  in  a  retrograde  or¬ 
der,  that  is  in  the  order  G,  F,  E,  D,  C,  B,  A,  G,  F 
&c.;  and  that  each  bissextile  has  two,  in  the  same 
order. 

It  is  now  usual  to  retain  only  the  dominical  letter  in 
the  calendar,  and  to  designate  the  other  days  of  the 
week  by  numbers,  or  by  their  names. 

14.  The  year  1800,  which  was  a  common  year, 
commenced  on  the  fourth  day  of  the  week,  and  conse¬ 
quently  the  dominical  letter  was  the  5th  of  the  alpha¬ 
bet,  w  hich  is  E.  From  thence,  taking  into  considera¬ 
tion,  that  every  four  years  in  which  a  bissextile  is  in¬ 
cluded,  requires  five  dominical  letters  in  a  retrograde 
order,  it  is  easy  to  find  the  dominical  letter  for  any 
year  in  the  present  century.  To  do  this,  multiply  the 


CHAPTER  XVI. 


211 


number  of  years  above  1S00.  by  5,  and  divide  the  pro¬ 
duct  by  4,  neglecting  the  remain  lee.  Divide  the  quo¬ 
tient  by  7>  and  subtract  the  remainder  from  5;  or  from 
12,  when  the  remainder  is  equal  to,  or  greater  than  5. 
The  last  remainder  is  the  number  of  the  dominical 
letter. 

Delambre,  in  the  38th  chapter  of  his  Astronomy  has 
given  the  investigation  of  a  formula  for  finding  the  do¬ 
minical  letter  in  any  century,  according  to  the  Gre¬ 
gorian  calendar. 

15.  There  are  some  periods  of  time,  which  though 
they  are  not  now  much  used,  it  may  be  proper  briefly 
to  notice. 

16.  The  Solar  Cycle  is  a  period  of  28  years,  in 
which,  according  to  the  Julian  calendar,  the  days  of 
the  week  return  to  the  same  days  of  the  month,  and  in 
the  same  order.  The  first  year  of  the  Christian  era 
was  the  10th  of  this  cycle.  Consequently  if  9  be  ad¬ 
ded  to  the  number  of  any  year,  and  the  sum  be  divided 
hy  28,  the  remainder  will  be  the  number  of  the  year 
of  the  solar  cycle.  When  there  is  no  remainder,  the 
year  is  the  28th  of  the  cycle. 

17*  The  Lunar  Cycle ,  or  as  it  is  sometimes  called, 
the  Metonic  Cycle,  is  a  period  of  19  years,  in  which 
the  conjunctions,  oppositions,  and  other  aspects  of  the 
moon,  return  on  the  same  days  of  the  year.  The  sy¬ 
nodic  revolution  of  the  moon  being  29.5305885  days, 
235  revolutions  are  6939.688  days;  which  differs  only 
about  an  hour  and  a  half  from  19  Julian  years.  The 
number  by  which  the  year  of  the  lunar  cycle  is  desig¬ 
nated,  is  frequently  called  the  Golden  Number. 

The  first  year  of  the  Christian  era  was  the  2d  of  the 
lunar  cycle.  Hence  to  find  the  year  of  the  cycle,  for 
any  given  year,  add  1  to  the  number  of  the  year,  and 


212 


CHAPTER  XVII. 


divide  by  19.  The  remainder  expresses  the  year  of 
the  cycle.  If  nothing  remains,  the  year  is  the  19th  of 
the  cycle. 

19.  The  Cycle  of  the  Indiction  is  a  period  of  15 
years.  This  period,  which  is  not  astronomical,  was 
introduced  at  Home,  under  the  emperors  and  had  re¬ 
ference  to  certain  judicial  acts. 

To  find  the  cycle  of  the  indiction  for  a  given  year, 
add  3,  and  divide  by  15.  The  remainder  expresses 
the  year  of  the  cycle. 

19.  The  Julian  Period  is  a  period  of  7980  years, 
obtained  by  taking  the  continued  product  of  the  num¬ 
bers  28 ,  19  and  15.  After  one  Julian  period  the  dif¬ 
ferent  cycles  of  the  sun,  moon  and  indiction,  return  in 
the  same  order,  so  as  to  be  just  the  same  iu  a  given 
year  of  the  period,  as  in  the  same  year  of  the  preceding 
period.  The  first  year  of  the  Christian  era  was  the 
4714th  of  the  Julian  period.  Hence  if  471*3  be  added 
to  the  number  of  a  given  year,  the  result  will  be  the 
year  of  the  Julian  period. 

20 .  The  Epact  as  an  astronomical  term  is  the  mean 
age  of  the  moon  at  the  commencement  of  a  year,  or  in 
other  words,  it  is  the  interval  between  the  commence¬ 
ment  of  the  year  and  the  time  of  the  last  mean  new 
moon;  and  is  expressed  in  days,  hours,  minutes  and 
seconds. 

21.  The  Epact,  as  given  in  the  calendar,  is  nearly 
the  age  of  the  moou  at  the  commencement  of  the  year, 
expressed  in  whole  days,  and  was  introduced  for  the 
purpose  of  finding  the  days  of  mean  new  and  full 
moon  throughout  the  year,  and  thence  the  times  of  cer¬ 
tain  festivals.  Without  entering  into  any  explanation 
of  the  reason  of  the  rule,  it  must  suffice  here  to  observe, 
that  the  Epact  for  any  year  during  the  present  century 


CHAPTER  XVII. 


213 


may  be  found  by  multiplying  the  golden  number  of  the 
year  by  11,  adding  19  to  the  product  and  dividing 
the  sum  by  30.  The  remainder  is  the  Epact  for  the 
year. 


CHAPTER  XVII. 

Universal  Gravitation  and  some  of  its  effects . 

1.  It  is  designed  to  give  in  this  chapter  a  general 
view  of  some  of  the  effects  of  the  attraction  of  gravita¬ 
tion,  without  entering  into  very  minute  investigations. 
The  propositions,  contained  in  the  first  four  of  the  fol¬ 
lowing  articles,  are  demonstrated  in  treatises  on  Me¬ 
chanics. 

2.  If  a  body  put  in  motion,  be  urged  towards  a 
fixed  point,  not  in  the  direction  of  its  motion,  by  a 
force  continually  acting  upon  it,  it  will  move  in  a  curve; 
and  the  straight  line  drawn  from  the  body  to  the  point, 
will  describe  areas  proportional  to  the  times. 

3.  Conversely,  if  a  body  move  in  a  curve,  in  such 
manner,  that  the  straight  line  drawn  from  it,  to  some 
point,  describes  areas  proportional  to  the  times,  the 
body  is  urged  towards  the  point  by  a  force  continually 
acting  on  it. 

By  Kepler’s  first  and  second  laws,  the  planets  re¬ 
volve  in  curves  about  the  sun,  and  their  radius  vectors 
describe  areas  proportional  to  the  times.  Consequently 
the  planets  are  urged  towards  the  sun  by  forces  con¬ 
tinually  acting  on  them. 

4.  If  a  body  revolving  about  a  point,  be  continually 
urged  towards  that  point,  by  a  force  which  varies  in¬ 
versely  as  the  square  of  the  distance,  it  will  move  in 


214 


ASTRONOMY. 


an  Ellipse  or  some  other  of  the  curves,  called  Conic 
Sections . 

5.  If  a  body  continually  urged  by  a  force,  directed 
to  some  point,  describe  an  ellipse  of  which  that  point 
is  a  focus,  the  force  must  vary  inversely  as  the  square 
of  the  distance. 

It  therefore  follows,  from  Kepler’s  second  law,  that 
each  planet  is  continually  urged  towards  the  sun,  by  a 
force  which  varies  inversely  as  the  square  of  the  dis¬ 
tance  from  the  sun’s  centre. 

6.  Since  each  planet  is  urged  towards  the  sun  by  a 
force,  varying  inversely  as  the  square  of  the  distance, 
it  is  reasonable  to  suppose,  instead  of  a  distinct  force 
for  each  planet,  a  single  force  residing  in  the  sun,  and 
varying  from  planet  to  planet  according  to  the  same 
law. 

7.  By  taking  into  view  Kepler’s  third  Iawr,  for  the 
motions  of  the  planets,  it  is  proved  that  the  sun  is  the 
centre  of  a  force,  which,  acting  on  the  particles  cf  mat¬ 
ter  in  all  the  planets,  and  varying  in  intensity,  inversely 
as  the  square  of  the  distance  from  the  sun’s  centre,  re¬ 
tains  them  in  their  orbits. 

8.  As  the  motions  of  the  satellites  of  Jupiter,  Saturn 
and  Uranus  are  conformable  to  Kepler’s  third  law,  it 
is  proved  in  like  manner  that  each  of  these  planets  is 
the  centre  of  a  force,  which  varying  in  intensity  in¬ 
versely  as  the  square  of  the  distance  from  the  centre  of 
the  planet,  extends  to  the  satellites  and  retains  them 
in  their  orbits. 

9.  The  earth  has  but  one  satellite,  and  therefore 
Kepler’s  third  law  does  not  apply  to  it.  But  by  in¬ 
vestigations,  founded  on  the  distance  which  a  heavy 
body  falls  at  the  earth’s  surface  in  one  second  of  time, 
compared  with  the  distance  which  the  moon  recedes 


CHAPTER  XVII. 


215 


in  the  same  time  from  a  tangent  to  its  orbit,  towards 
the  earth,  it  is  proved  that  the  force  of  gravity,  va¬ 
rying  inversely  as  the  square  of  the  distance,  extends 
to  the  moon  and  retains  it  in  its  orbit. 

10.  The  existence  of  a  similar  force,  in  each  of  the 
planets  that  have  no  satellites,  is  inferred  from  the  ef¬ 
fects  which  they  are  known  to  produce  on  one  another 
and  on  the  other  planets. 

41.  The  circumstances  mentioned  in  the  preceding 
articles  serve  to  prove  that  all  particles  of  matter  are 
urged  towards  one  another,  with  a  force  which  varies 
inversely  as  the  square  of  the  distance.  This  force  is 
called  the  Force  of  Gravitation. 

12.  A  Projectile  Force  is  the  force  by  which  a  body 
is  put  in  motion. 

13.  A  Centripetal  Force  is  the  force  by  which  a 
body  revolving  about  another  body  is  urged  towards  it. 

14.  A  Centrifugal  Force  is  the  force  by  which  a 
body  revolving  about  another  body,  tends  to  recede 
from  it. 

15.  Centripetal  and  Centrifugal  forces  are  called 
Central  Forces . 

Relative  Masses  of  the  Planets — Relative  weight  of  a 
body  at  their  surfaces. 

16.  The  relative  quantities  of  matter  or  masses  of 
the  sun,  planets,  and  satellites  may  be  determined  with 
considerable  accuracy,  from  the  effects  which  they 
produce  in  disturbing  the  motions  of  each  other.  For 
these  effects  depend  on  the  quantities  of  matter  of  the 
disturbing  bodies  and  on  their  distances;  and  the  dis¬ 
tances  are  known  from  the  methods  of  plane  astro¬ 
nomy. 

17.  The  masses  of  those  planets  which  have  satel- 


216  ASTRONOMY. 

lites  may  be  found  ina  simpler  manner  and  with  greater 
accuracy.  If  1  denote  the  mass  of  the  sun.  M  the 
mass  of  a  planet,  m  the  mass  of  one  of  its  satellites,  D 
the  mean  distance  of  the  planet  from  the  sun,  d  the 
mean  distance  of  the  satellite  from  the  planet,  and  P 
and  p  the  periodic  times  of  the  planet  and  satellite  re¬ 
spectively;  then  it  is  proved  that 

M  4-  m  d3  P2 
"l  r  M  "  D*  ~f 

As  the  mass  of  the  satellite  is  small  compared  with 
that  of  the  planet,  and  the  mass  of  the  planet  is  small 
compared  with  that  of  the  sun,  we  have, 

d3  P2 

M  ==  - ..  — very  nearly. 

D3  p2 

18.  The  following  table  exhibits  the  relative  quan¬ 
tities  of  matter  or  masses  of  the  sun  and  planets  as 
given  by  Laplace  in  the  fourth  edition  of  his  Systeme 
Du  Monde. 


Sun 

Mercury 

Venus 

The  Earth 

Mars 

Jupiter  - 

Saturn 

Uranus 


1 

1 

2025810 

1 

356632 

1 

337102 

1 

2546320 

1 

1066.09 

1 

3512.08 

1 

19504 


CHAPTER  XVII. 


247 


If  the  mass  of  the  earth  be  denoted  by  1,  the  mass 
of  the  moon,  according  to  the  most  accurate  deter¬ 
mination,  is  V£T. 

19.  The  densities  of  bodies  are  proportional  to  their 
quantities  of  matter,  divided  by  their  bulks.  The  fol¬ 
lowing  table  contains  the  densities  of  the  sun,  moon 
and  planets,  the  density  of  the  earth  being  denoted 
by  1. 


Sun 

0.252 

Mercury 

-  2.585 

Venus 

;  -  1.024 

The  Earth  - 

-  1.000 

The  Moon 

0.615 

Mars  - 

-  0.656 

Jupiter  - 

0.201 

Saturn 

-  0.103 

Uranus  - 

0218 

20.  Supposing  the  planets  to  be  exactly  spherical 
and  not  to  revolve  on  their  axes,  the  weight  of  the 
same  body  at  their  different  surfaces  would  be  pro¬ 
portional  to  their  quantities  of  matter,  divided  by  the 
squares  of  their  diameters.  But  the  centrifugal  force, 
at  the  surface  of  a  planet  that  revolves  on  its  axis,  di¬ 
minishes  the  weight  of  a  body,  placed  on  it,  particu¬ 
larly  near  the  equator.  The  diminution  thus  pro¬ 
duced,  on  any  of  the  planets,  is  not  however  very  con¬ 
siderable.  The  following  table,  taken  from  Vince’s 
Astronomy,  exhibits  the  relative  weight,  nearly,  of  a 
body  at  the  surface  of  the  sun  and  planets,  its  weight 
at  the  surface  of  the  earth  being  denoted  by  1. 

Sun  ,  27.70 

Mercury  -  1.70 

Venus  -  0.98 

29 


ASTRONOMY. 


SIS 


The  Earth  - 

1  00 

Mars 

0.34 

Jupiter  - 

2.33 

Saturn  - 

1.02 

Uranus  - 

0.93 

THE  CENTRE  OF  GRAVITY  OF  THE  SOLAR  SYSTEM. 

21.  As  all  particles  of  matter  attract  each  other,  the 
sun  must  be  attracted  towards  a  planet,  in  like  man¬ 
ner  as  the  planet  is,  towards  the  sun.  But  as  the 
quantity  of  matter  in  the  sun  is  far  greater  than  that  in 
any  of  the  planets,  its  attraction  at  a  given  distance 
must  be  proportion  ably  greater. 

22.  If  there  were  only  one  planet,  the  sun  and  that 
planet  would  describe  similar  ellipses,  of  which  their 
common  centre  of  gravity,  would  be  one  of  the  foci; 
their  distances  from  that  point,  being  always  inversely 
as  their  quantities  of  matter.  As  there  are  several 
planets  revolving  round  the  sun,  the  path  of  the  sun's 
centre  must  be  a  more  complicated  curve.  But  the 
quantity  of  matter  in  all  the  planets,  taken  together, 
being  very  small,  compared  with  that  in  the  sun,  the 
extent  of  the  curve  described  by  the  suits  centre  cau 
not  be  very  great. 

23.  It  is  found  by  computation,  that  the  distance  be¬ 
tween  the  sun’s  centre  and  the  centre  of  gravity  of  the 
system,  can  never  be  equal  to  the  sun’s  diameter. 

24.  It  is  proved  by  writers  on  Mechanics  that  the 
centre  of  gravity  of  a  system  of  bodies  is  not  affected 
b^7  the  mutual  actions  of  these  bodies  on  one  another; 
and  that  unless  there  are  extraneous  actions,  the  centre 
of  gravity  will  either  remain  at  rest  or  move  uniformly 
in  a  right  line 

35.  From  some  minute  changes  in  the  situations  of 


CHAPTER  XVII. 


219 


some  of  the  fixed  stars,  called  the  Proper  motions  of 
those  stars,  Dr.  Herschell  has  inferred  that  the  centre 
of  gravity,  and  consequently  the  whole  system,  of  the 
sun  and  planets,  is  in  motion  towards  the  constellation 
Hercules.  But  the  investigations  of  Dusejour  and 
Bnrckhardt  have  shown  that  the  observations,  hitherto 
made,  are  not  sufficient  to  prove  the  existence  of  any 
such  motion. 


kepler’s  laws. 


26.  Kepler’s  laws,  with  regard  to  the  motions  of 
the  planets,  have  been  thus  far  considered  as  rigor¬ 
ously  true.  It  may  now  be  proper  to  inform  the  stu¬ 
dent  that  the  mutual  actions  of  the  heavenly  bodies  on 
each  other,  cause  slight  deviations  from  those  laws,  as 
they  are  stated  in  the  preceding  part  of  the  work. 

27-  If  the  radius  vector  and  mean  distance  of  a  pla¬ 
net  be  reckoned  from  the  centre  of  gravity  of  the  sys¬ 
tem,  to  the  centre  of  the  planet,  or  when  the  planet  has 
satellites,  to  the  centre  of  gravity  of  the  planet  and 
satellites,  the  first  and  second  laws  will  hold  true,  ex¬ 
cepting  so  far  as  the  motion  of  the  planet  is  affected  by 
tbe  actions  of  the  others. 

28.  Tbe  third  law,  as  applied  to  any  two  of  the 
planets,  is  affected  not  only  by  the  actions  of  the  other 
planets,  but  also  by  the  quantities  of  matter  in  the  two 
planets  themselves.  If  p  and  P  be  the  periodic  revo¬ 
lutions  of  any  two  of  the  planets,  a  and  A  their  mean 
distances  from  the  centre  of  gravity  of  the  system,  and 
m  and  M  their  quantities  of  matter,  that  of  the  sun  be¬ 
ing  denoted  by  1,  then,  disregarding  the  actions  of  the 
other  planets, 


p2 :  P2 : 


1  +  m 


A3 

1  +  M 


220 


ASTRONOMY. 


PROBLEM  OF  THE  THREE  BODIES. 

29.  If  we  suppose  only  two  bodies  to  gravitate  to¬ 
wards  each  other,  with  forces  inversely  as  the  squares 
of  their  distances,  and  to  revolve  about  their  common 
centre  of  gravity,  they  would  move  in  conic  sections, 
and  the  radius  vectors  would  describe  areas  propor¬ 
tional  to  the  times;  the  centre  of  gravity  either  remain¬ 
ing  at  rest  or  moving  uniformly  in  a  right  line.  But  if 
there  are  three  bodies,  the  action  of  any  one  on  the 
other  two,  changes  the  nature  of  their  orbits,  so  that 
the  determination  of  tlieir  motions  becomes  a  problem 
of  the  greatest  difficulty,  distinguished  by  the  name  of 
The  Problem  of  the  Three  Bodies. 

30.  The  solution  of  the  problem  of  the  three  bodies, 
in  its  utmost  generality,  is  not  within  the  power  of  the 
mathematical  sciences,  as  they  now  exist.  Under  cer¬ 
tain  limitations,  however,  and  such  as  are  quite  con¬ 
sistent  with  the  condition  of  the  heavenly  bodies,  it 
admits  of  being  resolved.  These  limitations  are,  that 
the  force  which  one  of  these  bodies  exerts  on  the 
other  two,  is,  either  from  the  smallness  of  that  body, 
or  its  great  distance,  very  inconsiderable,  in  respect  of 
the  forces  which  these  two  exert  on  one  another. 

31.  The  force  of  the  third  body  is  called  a  disturb¬ 
ing  force,  and  its  effects  in  changing  the  places  of  the 
other  two  bodies  are  called  the  disturbances  of  the 
System. 

32.  Though  the  small  disturbing  forces  may  be  more 
than  one,  or  though  there  be  a  great  number  of  remote 
disturbing  bodies,  the  computation  of  tlieir  combined 
effect  arises  readily  from  knowing  the  effect  of  one; 
and  therefore  the  problem  of  three  bodies,  under  the 
conditions  just  stated,  may  be  extended  to  any  number. 


CHAPTER  XVII. 


£21 


33.  The  problem  of  the  three  bodies  has  exercised 
the  ingenuity  of  several  of  the  most  eminent  mathema¬ 
ticians.  But  Laplace,  in  the  Meeanique  Celeste,  has 
extended  the  solution  farther  than  any  other  person. 
He  has  given  a  very  complete  investigation  of  the  ine¬ 
qualities,  both  of  the  planets  and  satellites. 

INEQUALITIES  OF  THE  MOON. 

34.  The  moon  is  attracted  at  the  same  time  by  both 
the  earth  and  sun;  it  is  only,  however,  the  difference 
between  the  gravitations  of  the  earth  and  moon  to¬ 
wards  the  sun  that  disturbs  the  motion  of  the  moon  about 
the  earth.  If  the  sun  were  at  an  infinite  distance,  they 
would  be  attracted  equally,  and  in  parallel  straight 
lines;  and  in  that  case  their  relative  motions  would  not 
be  in  the  least  disturbed.  But  his  distance,  although 
very  great  in  respect  of  that  of  the  moon,  yet  can  not  be 
supposed  infinite;  the  moon  is  alternately  nearer  the 
sun,  and  farther  from  him  than  the  earth,  and  the 
straight  line  which  joins  her  centre  and  that  of  the 
sun,  forms  with  the  terrestrial  radius  vector  an  angle 
which  is  continually  varying;  thus  the  sun  acts  une¬ 
qually  and  in  different  directions  on  the  earth  and 
moon,  and  from  this  diversity  of  action  there  must  re¬ 
sult  inequalities  in  her  motion  which  depend  on  her 
position  in  respect  of  the  sun. 

35.  At  the  quadratures,  the  gravity  of  the  moon  to 
the  earth  is  increased  in  consequence  of  the  sun’s  ac¬ 
tion,  by  a  quantity  equal  to  the  product  of  the  mass  of 
the  sun,  by  the  distance  of  the  moon  from  the  earth, 
divided  by  the  cube  of  the  earth’s  distance  from  the 
sun;  at  the  syzigies  it  is  diminished  by  twice  this 
quantity;  and  the  effect  in  the  whole,  is  a  diminution 
of  the  moon’s  gravity,  equal  to  the  product  of  the  sun’s 


ASTRONOMY. 


mass  by  the  moon’s  mean  distance  from  the  earfh,  di¬ 
vided  by  twice  the  cube  of  the  earth’s  distance  from 
the  sun.  And  the  value  of  the  mean  diminution  is 
equal  to  a  358th  part  of  the  whole  gravity  of  the  moon 
to  the  earth. 

It  is  a  well  known  proposition  in  Mechanics,  that  if  AB  and 
AD,  Fig,  50,  represent  the  quantities  and  directions  of  two  forces 
acting  on  a  point  or  body  at  A,  and  the  parallelogram  ABCD  be 
completed,  the  diagonal  AC  will  represent  the  quantity  and  di¬ 
rection  of  a  single  force  which  would  produce  the  same  effect  as 
the  two  forces.  The  substitution  of  a  single  force  as  the  equiva¬ 
lent  of  two  others,  is  called  the  Composition  of  Forces. 

On  the  contrary  if  AC,  represent  the  quantity  and  direction  of 
a  single  force  acting  on  a  body  at  A,  and  any  parallelogram  ABCD 
is  described  about  AC  as  a  diagonal,  the  adjacent  sides  AB  and 
AD  will  represent  the  quantities  and  directions  of  two  forces  that 
are  just  equivalent  to  the  single  force.  The  substitution  of  two 
forces,  as  the  equivalent  of  a  single  force,  is  called  the  Resolution 
of  Forces.  » 

Let  ACBO,  Fig.  51,  represent  the  orbit  of  the  moon  which 
may  in  this  investigation  be  considered,  as  coinciding  with  the 
plane  of  the  ecliptic.  Also  let  S  be  the  sun,  E  the  earth,  M  the 
place  of  the  moon  in  her  orbit,  and  AB,  perpendicular  to  SE,  the 
line  of  the  quadratures.  Let  the  line  SE  represent  the  force 
which  the  sun  exerts  on  the  earth  at  E  or  on  the  moon,  when  in 

ST3 

quadratures,  at  A  and  B*.  Then,  SM2 :  SE2 :  SE  :  =  the 

5i\r 

force  with  which  the  sun  acts  on  the  moon  at  M.  In  the  line 

SE3 

MS,  produced  if  necessary,  take  MD  =  _ — ;  then  MD  repre- 

SM2 

sents  the  force  which  the  sun  exerts  on  the  moon  at  M.  Let  the 

*  Strictly  speaking,  as  the  quantity  of  matter  in  the  earth  is  greater  than 
that  in  the  moon,  the  forces  which  the  sun  exerts  on  the  earth  and  moon 
when  at  equal  distances,  are  not  equal.  But  the  effects  of  those  forces,  in 
moving  the  bodies,  are  equal,  and  it  is  these  effects,  which  is  the  subject 
under  consideration. 


CHAPTER  XVII. 


223 


force  MD  be  resolved  into  the  two  MH  and  MG,  one  of  which, 
MH  is  equal  and  parallel  to  ES.  Then  since  the  force  MH  is 
equal  and  parallel  to  ES,  it  will  have  no  tendency  to  change  the 
relative  motions  or  positions  of  the  earth  and  moon.  The  other 
force  MG,  will  therefore  represent,  in  quantity  and  direction,  the 
whole  effect  of  the  sun’s  action  in  disturbing  the  moon’s  motion 
in  her  orbit.  Let  SM  be  produced  to  meet  the  diameter  AB  in 
N.  Then  because  the  angle  ESN  is  very  small,  being  when 
greatest  only  about  7',  the  line,  SN  may  be  considered  equal 
to  SE.  Hence, 

mf>  _  SE3  _  SN3  -  (SM  +  MN')3 
SM2  SM2  SM2 

_  SM3  -f  3  SM2  x  MN  +  3  SM  x  MN2  +  MN3 
SM2 


But  as  MN  is  very  small  compared  with  SM,  the  two  terms 
3SM  x  MN2,  and  MN5  may  be  omitted.  Therefore, 


MD 


=  SM3  +  3  SM2  x  MN  _  gM  +  3  MN 
SM2 


Or,  SD  =  3  MN. 

As  the  angle  ESM  is  very  small,  and  SD  is  also  small,  the  line 
DG  must  very  nearly  coincide  with  SE,  and  consequently  the 
point  G  with  the  point  L.  We  may  therefore  consider  ML  as 
the  force  by  which  the  sun  disturbs  the  motion  of  the  moon. 
Now, 

EL  +  LS  =  ES  =  HM  =  DG  =SD  +LS,  very  nearly, 
or,  EL  =  SD,  very  nearly. 

Hence,  if  MK  be  perpendicular  to  SE,  we  have, 

EL  =  3  MN  =  3  EK. 


Let  the  force  of  ML  be  resolved  into  two  others,  one  MQ,  in 
the  direction  of  the  radius  vector,  and  the  other  MP  in  the  direc¬ 
tion  of  a  tangent  to  the  orbit  at  M.  Then  the  force  MQ  increases  or 
diminishes  the  gravity  of  the  moon  to  the  earth,  according  as  the 
point  Q  falls  between  E  and  M,  or  in  EM  produced.  The 
other  force  MP  increases  or  diminishes  the  moon’s  angular  mo- 


ASTRONOMY. 


224 


lion  about  the  earth.  Since  the  moon’s  orbit  does  not  differ  much 
from  a  circle,  the  angle  QMP  may  be  considered  as  a  right  an¬ 
gle.  Put  a  =  SE,  r  =  EM,  and  x  =  the  angle  AEM.  Then, 

EK  =  EM  cos  MEK  r  sin  x 
EL  —  3  EK  =  3  r  sin  x , 

3  r 

PM  as  LQ  =  EL  cos  x  —  3  r  sin  x  cos  x  =  —  sin  2  x,  ( App.  7). 

2 


Also,  EQ  =  EL  sin  x  —  Sr  sin2  x, 

MQ  =  EQ  —  EM  =  3  r  sin2  x  —  r  =  —  r  ( 1  —  3  sin2  x)* 

Or  using  the  affirmative  sign  to  denote  an  increase  in  the 
moon’s  gravity  to  the  earth, 

MQ  -f  r  1  —  3  sin2  x). 

Now  if  m  —  the  mass  of  the  sun,  then  the  force  which  the  sun 

TYl 

exerts  on  the  earth  may  be  expressed  by  — .  Hence, 


ES  :  PM  :  : 

the  force  PM  =  — . 

a2 


—  :  the  force  PM.  Therefore, 
a2 

PM 

ES 


m.3r  sin  2x 
2a3 


3mr 

Jo3" 


sin2x 


A. 


In  like  manner, 

the  force  MQ  —  — .  MS  —  JHHL.  (I  — 3  sin2  x).  B 

a2  ES  a3 

When  the  moon  is  in  quadratures,  x  =  0  or  180°,  and  conse¬ 
quently. 


The  force  MQ  =  +  —. 

a 3 


When  the  moon  is  in  syzigies,  x  =  90°  or  270°,  and, 
therefore, 

The  foree  MQ  =  —  ^HOL. 

a3 

The  force  MQ  is  =  0,  when  3  sin*  x  =  1,  or  sin  x  =  y/  -J; 
that  is,  when  x  —  35°  15'  52". 


The  moon’s  gravity  to  the  earth  is  therefore  increased  while 
she  is  within  about  35°  of  the  quadratures,  on  either  side,  and  is 


CHAPTER  XVIT. 


225 


diminished  in  all  the  remaining  part  of  the  orbit;  and  the  greatest 
diminution  is  double  the  greatest  increase.  It  follows  therefore 
that  in  the  whole,  the  moon’s  gravity  to  the  earth  is  diminished  by 
the  action  of  the  sun.  A  short  fluxional  investigation  proves  that 

the  mean  diminution  is  ;  r  representing  in  this  case  the  mean 

distance  of  the  moon  from  the  earth.  And  it  has  been  found  that 
the  value  of  this  expression  is  equal  to  the  358th  part  of  the 
whole  gravity  of  the  moon  to  the  earth. 

36.  From  the  diminution  of  her  gravity  by  a  358th 
part,  the  moon  describes  her  orbit  at  a  greater  distance 
from  the  earth,  with  a  less  angular  velocity,  and  in  a 
longer  time,  than  if  she  were  acted  on,  only  by  the 
attraction  of  the  earth. 

3/.  The  inequality  in  the  moon’s  motion,  called  the 
Annual  Equation,  (IO.&9),  is  the  effect  of  the  varia¬ 
tion  in  the  distance  of  the  earth  from  the  sun. 


Since,  in  the  expression  JlilL.,  which  designates  the  mean  di- 

2(i  ^ 

minution  in  the  moon’s  gravity  to  the  earth,  the  quantities  m  and  r, 
are  constant,  it  follows,  that  the  mean  diminution  is  inversely  pro¬ 
portional  to  the  cube  of  the  earth’s  distance  from  the  sun.  Hence 
as  the  earth  approaches  the  perihelion,  its  distance  diminishing,  the 
mean  diminution  of  the  moon’s  gravity  to  the  earth  must  increase; 
the  moon’s  distance  from  the  earth  must  become  greater  than  it 
otherwise  would  be;  and  consequently  its  motion  must  be  slower. 
The  contrary  takes  place  as  the  earth  approaches  the  aphelion. 


38,  The  Evection  is  produced  by  an  inequality  in  the 
sun’s  disturbing  force,  depending  011  the  variation  in 
the  moon’s  distance  from  the  earth,  and  on  the  posi¬ 
tion  of  the  moon  with  respect  to  the  line  of  the 
syzigies. 


Let  R  and  r  denote  the  distances  of  the  moon  from  t\\e  earth, 

SO 


226 


ASTRONOMY. 


in  apogee  and  perigee,  when  the  line  of  the  apsides  coincides  with 
the  line  of  the  sjzigies,  X  and  x,  the  distances  at  which  the  moon 
would  be  from  the  earth,  in  apogee  and  perigee,  if  she  was  not 
acted  on  by  the  sun,  and  G  and  g  the  perigean  and  apogean  gra¬ 
vities  in  that  case.  Also  put  n  =  and  supposing  the  earth’s 

distance  from  the  sun  to  remain  constant,  n  will  be  constant. 
Then  (35),  G  —  2 rn  and  g  —  2R n,  will  be  the  perigean  and 
apogean  gravities  of  the  moon,  when  the  line  of  the  apsides 
coincides  with  the  line  of  the  syzigies.  Hence, 

X2  :  x2  :  ’  G  :  g, 

and  R2  :  r2 :  :  G  —  2rn  :  g  —  2R n. 

Consequently, 

X2  =  G 

*2  g' 

and  5!  = 

r2  g —  2Rn 

Now  as  G  is  greater  than  g>  and  2rn,  less  than  2Rn,  it  is  evi¬ 
dent  that, 

is  greater  than  — . 

g  -  2Rn  8  g 

R2  #  %2 

Hence,  - —  is  greater  than  — , 
r 2  x 2 

It  therefore  follows,  that  when  the  line  of  the  apsides  coincides 
with  the  line  of  the  syzigies,  the  ratio  of  the  apogean  distance  of 
the  moon  to  the  perigean  distance,  and  consequently  the  eccen¬ 
tricity  of  the  orbit,  is  increased  by  the  action  of  the  sun.  In  like 
manner  it  may  be  shown  that  when  the  line  of  the  apsides 
coincides  with  the  line  of  the  quadratures,  the  sun’s  action  di¬ 
minishes  the  eccentricity  of  the  orbit.  The  change  in  the  ec¬ 
centricity  of  the  orbit  produces  a  change  in  the  equation  of  the 
centre;  which  change  is  the  evection. 

39.  The  Variation  is  produced  by  a  part  of  the  sun’s 
disturbing  force,  which  acts  in  the  direction  of  a  tan 
gent  to  the  moon’s  orbit. 


I 


CHAPTER  XVII.  227 

It  has  been  shown  (34.A),  that  MP,  the  part  of  the  sun’s 
force  which  acts  in  the  direction  of  a  tangent  to  the  orbit,  is  equal 

to  sin  2x.  Hence,  supposing  the  earth’s  distance  from 

the  sun,  and  moon’s  distance  from  the  earth  to  remain  constant, 
this  force  is  proportional  to  sin  2x ;  that  is,  to  the  sine  of  twice  the 
distance  of  the  moon  from  the  quadratures.  It  is  therefore  great¬ 
est  in  the  octants;  and  nothing  in  the  syzigies  and  quadratures. 

Supposing  the  moon  to  set  out  from  the  quadrature  A,  the  tan¬ 
gential  force  MP  continually  accelerates  her  motion,  till  she  ar¬ 
rives  at  the  syzigy  C;  the  force  then  changes  its  direction  and 
retards  her  motion.  Consequently  at  C  the  motion  is  greatest. 
As  the  moon  advances  from  C,  her  motion  is  continually  re¬ 
tarded  till  she  arrives  at  B,  where  it  is  least.  It  is  then  accele¬ 
rated  till  it  becomes  greatest  at  0,  and  again  retarded  till  it  be¬ 
comes  least  at  A.  Hence,  as  the  motion  is  greatest  in  the  syzigies 
and  least  in  the  quadratures,  and  as  the  degree  of  retardation  is  the 
same  as  that  of  acceleration,  we  may  infer  that  the  mean  motion* 
has  place  when  the  moon  is  in  the  octants. 

Now  as  the  moon  moves  from  the  quadrature  A  with  a  motion 
less  than  her  mean  motion,  her  mean  place  will  be  in  advance  of 
her  true  place,  and  will  become  more  and  more  so,  till  at  the  oc¬ 
tant,  the  true  motion  is  equal  to  the  mean.  The  difference  be¬ 
tween  the  true  and  mean  places  is  then  the  greatest.  For  after 
that,  the  true  motion  being  greater  than  the  mean,  the  true  place 
will  approach  nearer  to  the  mean,  till  at  the  syzigy  C,  they 
coincide.  It  is  equally  plain,  that  at  the  octant  between  C  and 
B,  the  moon’s  true  place  will  be  most  in  advance  of  the  mean 
place,  and  that  at  B,  they  will  again  coincide.  Corresponding 
effects  take  place  in  the  two  remaining  quadrants. 

40.  The  inequality  called  the  Acceleration  of  the 
Moon  (10.21),  by  which  her  velocity  appears  subject  to 
continual  increase,  and  her  period  to  continual  diini- 

*  The  expressions,  mean  place,  true  place,  mean  motion  and  true  motion, 
are  hereto  be  understood,  only  in  relation  to  the  present  inequality. 


228 


ASTRONOMY. 


notion,  lias  been  found  by  Laplace  to  be  a  Secular 
equation,  depending  on  a  change  in  the  eccentricity  of 
the  earth’s  orbit,  produced  by  the  actions  of  the  planets, 
and  which  requires  several  thousand  years  to  go 
through  its  different  values. 

MOTION  OF  THE  APSIDES  OF  THE  MOON’S  ORBIT. 

4fl.  The  motion  of  the  apsides  is  produced  by  the 
action  of  the  sun,  in  diminishing  the  moon’s  gravity  to 
the  earth. 

If  the  moon  was  only  acted  on  by  the  earth’s  attraction,  it 
would  describe  an  ellipse,  and  its  angular  motion  would  be  just 
180°,  from  one  apsis  to  the  other;  or  which  is  the  same,  from  one 
place  where  the  orbit  cuts  the  radius  vector  at  right  angles,  to  the 
other.  But  in  consequence  of  the  change  produced  in  the  moon’s 
gravity  to  the  earth,  by  the  action  of  the  sun,  the  moon’s  path  is 
not  an  ellipse.  When  the  effect  of  the  sun’s  action  is  a  diminu¬ 
tion  of  the  moon’s  gravity,  she  will  continually  recede  from  the 
ellipse  that  would  otherwise  be  described,  her  path  will  be  less 
bent,  and  she  must  move  through  a  greater  distance  before -the 
radius  vector  intesects  the  path  at  right  angles.  She  must  there¬ 
fore  move  through  a  greater  angular  distance  than  180°,  in  going 
from  one  apsis  to  the  other,  and  consequently  the  apsides  will  ad¬ 
vance.  On  the  contrary,  when  the  gravity  is  increased  by  the 
sun’s  action,  the  moon’s  path  will  fall  within  the  ellipse  which  she 
would  otherwise  describe,  its  curvature  will  be  increased,  and  the 
distance  through  which  she  must  move  before  the  radius  vector  in¬ 
tersects  her  path  at  right  angles,  will  be  less.  The  apsides  will 
therefore  move  backwards.  Now  it  has  been  shown  (35)  that 
the  sun’s  action,  alternately  diminishes  and  increases  the  moon’s 
gravity  to  the  earth.  The  motion  of  the  apsides  will  therefore 
be  alternately  direct  and  retrograde.  But  as  the  diminution  has 
place  during  a  much  longer  part  of  the  moon’s  revolution,  and  is 
besides  greater  than  the  increase,  the  direct  motion  will  exceed 
the  retrograde.  Consequently  in  an  entire  revolution  of  the 
moon,  the  apsides  have  a  progressive  motion. 


CHAPTER  XVII. 


£29 


MOTION  OF  THE  MOON’S  NODES,  AND  CHANGE  IN  THE 
INCLINATION  OF  THE  ORBIT. 

42.  The  direction,  in  which  the  sun’s  disturbing 
force  acts  on  the  moon,  does  not,  except  in  some  par¬ 
ticular  cases,  coincide  with  the  plane  of  the  moon’s  orbit; 
this  force  therefore  produces  a  tendency  in  the  moon 
to  quit  that  plane,  one  of  the  effects  of  which,  i9  a 
change  in  the  position  of  the  line  of  the  nodes;  and 
another,  is  a  change  in  the  inclination  of  the  plane  of 
the  orbit  to  that  of  the  ecliptic. 

Let  OL,  Fig.  52,  be  the  line  passing  through  the  centres  of 
the  earth  and  sun,  and  IN'  the  line  of  the  nodes.  These  two  lines 
lie  in  the  plane  of  the  ecliptic,  which  we  may  consider  as  desig¬ 
nated  by  the  plane  of  the  paper.  Let  EMHI  conceived  to  be, 
from  El,  above  the  plane  of  the  paper,  be  the  plane  of  the 
moon’s  orbit,  NM  a  part  of  the  northern  half  of  the  orbit,  and 
AB  a  plane,  seen  edgewise,  perpendicular  to  the  line  EL.  When 
the  moon  is  in  this  lattter  place  it  is  in  quadrature. 

Let  ML  designate  the  quantity  and  direction  of  the  sun’s  dis¬ 
turbing  force  when  the  moon  is  at  M.  Now  when  the  line  of  the 
nodes  coincides  with  OL  the  line  of  the  syzigies,  ML  will 
coincide  with  the  plane  of  the  moon’s  orbit,  and  will  therefore 
have  no  tendency  to  make  the  moon  deviate  from  that  plane. 
Also,  since  EL  is  equal  to  three  times  the  distance  of  the  moon 
from  AB  (34),  when  the  moon  is  in  the  plane  AB,  that  is  when 
she  is  in  quadrature,  L  will  coincide  with  E,  and  consequently 
ML  will  be  in  the  plane  of  the  orbit,  and  will  have  no  tendency 
to  make  the  moon  move  from  it.  At  all  other  times,  the  force 
ML,  not  acting  in  the  plane  of  the  orbit,  will  tend  to  make  the 
moon  quit  that  plane;  or  instead  of  supposing  the  moon  continually 
to  pass  from  one  plane  to  another,  we  may  conceive  the  plane  it¬ 
self  to  change  its  position. 

Let  LH  be  drawn  perpendicular  to  IHME  the  plane  of  the 
moon’s  orbit.  Then  if  MH  be  joined,  and  the  parallelogram 
MHLK  be  completed,  the  lines  MH  and  MK  will  represent,  in 


230 


ASTRONOMY. 


quantities  and  directions,  two  forces  that  are  together  equivalent 
to  ML.  The  force  MH  acting  in  the  plane  of  the  orbit,  has  no 
tendency  to  change  the  position  of  that  plane.  The  tendency  of 
the  other  force  MK,  acting  at  right  angles  to  the  plane  of  the  or¬ 
bit,  will  be  to  bend  the  moon’s  path  towards  the  ecliptic,  or  from  it. 
When  the  effect  of  the  force  MK  is  to  bend  the  moon’s  path  to¬ 
wards  the  ecliptic,  the  moon  will  meet  the  ecliptic  sooner  than  it 
would  otherwise  do,  and  consequently  the  node  will  move  back¬ 
wards.  On  the  contrary,  when  the  force  MK  bends  the  moon’s 
path  from  the  ecliptic,  the  moon  will  not  meet  the  ecliptic  so  soon 
as  it  would  otherwise  do,  and  therefore  the  node  will  move  for¬ 
ward,  Now  it  is  plain  that  when  the  points  L  and  M  are  on  the 
same  side  of  the  line  of  the  nodes,  the  force  MK  tends  to  make 
the  moon’s  path  bend  towards  the  ecliptic;  and  when  they  are 
on  opposite  sides,  it  tends  to  make  the  path  bend  from  the  ecliptic. 
Hence  when  the  points  L  and  M  are  on  the  same  side  of  the  line 
of  the  nodes,  the  motion  of  the  nodes  is  retrograde;  and  when  on 
opposite  sides,  it  is  direct. 

When  the  line  of  the  nodes  has  the  position  NN  the  points  L 
and  M  will  be  on  the  same  side  of  it,  while  the  moon  is  moving 
from  the  node  N  to  the  next  quadrature  in  EB;  and  therefore  the 
motion  of  the  nodes  is  retrograde.  When  the  moon  has  passed 
the  quadrature,  the  point  L  falls  on  the  other  side  of  E,  in  EO; 
and  therefore  while  the  moon  is  moving  from  the  quadrature  to  the 
next  node  in  EN',  the  point,  L  and  M  will  be  on  opposite  sides  of 
the  line  of  the  nodes,  and  the  motion  of  the  nodes  will  be  direct. 
While  the  moon  is  moving  from  the  node  in  EN'  to  the  quad¬ 
rature  in  EA,  the  motion  of  the  nodes  will  be  again  retrograde; 
and  while  she  is  moving  from  the  quadrature  in  EA,  to  the  node 
in  El,  it  will  be  direct.  Hence,  while  the  moon  is  moving  from 
the  nodes  to  the  quadratures,  the  motion  of  the  nodes  is  retro¬ 
grade;  and  while  she  is  moving  from  the  quadratures  to  the  nodes, 
it  is  direct.  It  is  therefore  plain  that  the  retrograde  motion  has 
place  during  a  longer  portion  of  the  moon’s  synodic  revolution, 
than  the  direct  motion. 

When  the  line  of  the  nodes  has  the  position  nn'  it  is  easy  to 
determine  from  what  has  been  said,  that  the  motion  of  the  nodes 


CHAPTER  XVII. 


231 

will  be  direct  while  the  moon  is  moving  from  the  nodes  to  the 
quadratures;  but  retrograde  while  she  is  moving  from  the  quad¬ 
ratures  to  the  nodes;  and  therefore,  that,  in  the  whole  synodic  re¬ 
volution  of  the  moon,  the  retrograde  motion  has  place  during  a 
longer  time  than  the  direct  motion. 

It  appears  then  that  in  each  synodic  revolution  of  the  moon, 
the  nodes  alternately  retreat  and  advance,  but  that  in  all  cases, 
except  when  the  line  of  the  nodes  nearly  coincides  with  the  line 
of  the  syzigies,  the  motion  is  retrograde  during  a  longer  time 
than  it  is  direct. 

Let  the  plane  LIH  be  perpendicular  to  IN  the  line  of  the 
nodes.  Then  the  angle  LIH  is  the  inclination  of  the  plane  of 
the  moon’s  orbit  to  the  ecliptic.  As  LH  is  perpendicular  to  the 
plane  of  the  orbit,  the  angle  IHL  is  a  right  angle.  Put  x  =  the 
moon’s  angular  distance  from  the  quadratures,  S  =  IEL  the  an¬ 
gle  contained  between  the  line  of  the  nodes  and  the  line  of  the 
syzigies,  I  =  LIH  the  inclination  of  the  orbit,  and  r  =  EM  == 
radius  vector  of  the  moon.  Then  (35),  EL  =  3r  sin  x.  Hence, 


LI  =  EL  sin  LEI  =  3r  sin  x  sin  S, 

MK  =  LH  =  LI  sin  LIH  ==  3r  sin  x  sin  S  sin  I, 

HI 

Or,  (35),  using  —  to  denote  the  force  exerted  by  the  sun  on  the 
a 2 

earth, 

m,  r  A/rTr  m.MK  3mr  .  .  a  .  , 

1  he  force  MK  =■  -  ==  - sm  x  sin  S  sin  I. 

a3  a 3 

Now  during  any  one  revolution  of  the  moon,  none  of  the  quan¬ 
tities  which  enter  into  the  expression  for  the  force  MK,  varies 
much,  except  sin  x.  And  it  is  easy  to  perceive,  by  reference  to 
the  figure,  that  sin  #,  and  consequently  the  force  MK,  always  ac¬ 
quires  its  greatest  value,  during  the  time  the  motion  of  the  nodes 
is  retrograde. 

As  in  each  synodic  revolution  of  the  moon,  the  nodes  retreat 
during  a  longer  time  than  they  advance,  and  as  the  force  which 
causes  the  motion  is  greatest  while  they  retreat,  the  retrograde 
motion  must  exceed  the  direct  motion,  and  the  result  in  the  whole 
must  be  a  retrograde  movement  of  the  nodes. 


astkoxomV. 


232 

When  the  tendency  of  the  force  MK  is  to  bend  the  moon’s 
path  towards  the  ecliptic,  if  the  moon  is  then  moving  from  the 
node  to  the  90°  from  it,  the  inclination  of  the  orbit  will  be  di¬ 
minished;  but  if  she  is  moving  from  the  90°  to  the  node,  the  in¬ 
clination  will  be  increased.  On  the  contrary,  when  the  tendency 
of  the  disturbing  force  is  to  bend  the  path  from  the  ecliptic,  the 
inclination  of  the  orbit  will  be  increased  when  the  moon  is  moving 
in  the  first  90°  from  the  node,  and  will  be  diminished  when  she 
is  moving  from  the  90°  to  the  node.  Hence  when  NN'  is  the 
line  of  the  nodes,  if  the  moon  set  out  from  the  quadrature  in  EA, 
the  inclination  of  the  orbit  will  be  continually  diminished  till  she 
is  90°  past  the  node  N;  and  will  then  be  increased  till  she 
arrives  at  the  quadrature  in  EB;  from  thence  to  the  90°  past  the 
node  in  EN',the  inclination  will  be  again  diminished,  and  will  then 
be  increased  till  she  again  arrives  at  the  quadrature  in  EA.  The 
diminution  will  therefore  be  greater  than  the  increase.  But  when 
nn'  is  the  line  of  the  nodes,  if  the  moon  set  out  from  the  quadra¬ 
ture  in  EA,  the  inclination  will  only  be  diminished  till  she  ar¬ 
rives  at  the  90°  from  the  node  in  E n',  and  will  be  increased  from 
thence  to  the  quadrature  in  EB;  it  will  then  be  diminished,  till 
she  is  90°  from  the  node  in  Ew,  and  will  be  increased  from  thence 
till  she  returns  to  the  quadrature  in  EA.  The  increase  will 
therefore  exceed  the  diminution.  Thus  in  some  synodic  revolu¬ 
tions  of  the  moon  the  inclination  of  the  orbit  is  diminished,  and 
in  others  it  is  increased  as  much.  The  result  is  a  mean  inclina¬ 
tion  which  does  not  change. 

43.  Disturbances  in  the  motions  of  the  earth  and 
planets  are  necessary  effects  of  the  actions  of  theses 
bodies  on  one  another;  but  it  is  not  designed  to  take  any 
other  notice  of  them  here,  than  to  mention  one  impor¬ 
tant  fact. 

44.  Lagrange  and  Laplace  have  proved  that  no 
terms  only  those  which  alternately  increase  and  di¬ 
minish,  can  enter  into  the  expressions  for  the  disturb¬ 
ances  of  the  planets.  This  proves  that  the  system  is 


CHAPTER  XVII. 


233 


stable;  that  it  does  not  involve  any  principle  of  de¬ 
struction  in  itself,  but  4s  calculated  to  endure  for 
ever,  unless  the  action  of  an  external  power  is  intro¬ 
duced. 

FIGURE  OF  THE  EARTH. 

45.  It  has  already  been  inferred  from  observation 
(4.8)  that  the  figure  of  the  earth  is  an  oblate  spheroid, 
of  which  the  greater  axis,  that  is,  the  diameter  of  the 
equator  is  to  the  less,  the  axis  of  revolution  as  321  to 
320. 

46.  Since  the  earth  revolves  on  its  axis,  it  is  evi¬ 
dent,  that  its  parts  are  all  under  the  influence  of  a  cen¬ 
trifugal  force,  varying  with  their  distances  from  that 
axis,  and  that  if  the  whole  were  a  fluid  mass,  the 
columns  towards  the  equator,  being  composed  of  parts 
that,  having  a  greater  centrifugal  force,  tend  more  to 
recede  from  the  axis,  must  extend  in  length,  in  order 
to  balance  the  columns  in  the  direction  of  the  axis.  By 
this  means  an  oblateness  or  elevation  at  the  equator 
would  be  produced,  similar,  in  some  degree  at  least,  to 
that  which  the  earth  has  been  found  to  possess. 

47.  A  homogeneous  fluid  of  the  same  mean  density 
with  the  earth,  and  revolving  on  its  axis  in  the  same 
time  that  the  earth  does,  would  be  in  equilibrium,  if  it 
had  the  figure  of  an  oblate  spheroid,  of  which  the 
axis  was  to  the  equatorial  diameter  as  229  to  230. 

48.  If  the  fluid  mass,  supposed  to  revolve  on  its 
axis,  be  not  homogeneous,  but  be  composed  of  strata 
that  increase  in  density  towards  the  centre;  the  solid 
of  equilibrium  will  still  be  an  elliptic  spheroid,  but  of 
less  oblateness  than  if  it  were  homogeneous. 

49.  Hence  as  the  ellipticity  of  the  earth  is  less  than 
being  about  7  it  is  evident,  that  if  the  earth  is 

31 


234 


ASTRONOMY. 


a  spheroid  of  equilibrium;  it  is  denser  toward  the  in¬ 
terior. 

50.  The  greater  density  of  the  earth  towards  the 
centre  has  been  proved  by  very  accurate  observations 
made  on  the  sides  of  the  mountain  Schehallier,  in 
Scotland,  by  Dr.  Maskelyne.  From  the  effect  of  the 
mountain  in  changing  the  direction  of  a  plumb  line 
suspended  near  it;  and  from  the  known  figure  and 
bulk  of  the  mountain  determined  by  a  survey,  it  was 
found  that  the  mean  density  of  the  mountain  w  as  to  the 
mean  density  of  the  earth  nearly  as  5  to  9. 

51.  The  inequalities  on  the  surface  of  the  earth,  and 
the  unequal  distribution  of  the  rocks  which  compose  it, 
with  respect  to  density,  must  produce  great  local  ir¬ 
regularities  in  the  direction  of  the  plumb  line,  and  are 
probably  in  part  the  causes  of  the  inequalities  observed 
in  the  measurement  of  contiguous  arches  of  the  meri¬ 
dian,  even  when  the  work  has  been  conducted  with 
the  greatest  skill  and  accuracy.  These  irregularities 
are  so  considerable  that  the  ellipticity  of  the  spheroid 
which  agrees  best  with  the  measurement  of  some  de¬ 
grees,  is  nearly  double  what  may  be  accounted  the 
mean  ellipticity. 

52.  From  accurate  observations  of  the  lengths  of 
pendulums  oscillating  seconds  at  places  in  different 
latitudes,  the  relative  force  of  gravity  at  the  places 
may  be  determined  and  from  thence  the  ellipticity  of 
the  earth. 

PRECESSION  OF  THE  EQUINOXES  AND  NUTATION  OF 
THE  EARTH’S  AXIS. 

53.  The  physical  investigation  of  the  precession  of 
the  equinoxes  is  a  subject  of  considerable  difficulty.  It 


CHAPTER  XVII. 


235 


must  suffice  here,  just  to  state  that  the  precession  is 
produced  by  the  actions  of  the  sun  and  moon  on  those 
parts  of  the  eartli  which  are  on  the  outside  of  a  sphere, 
conceived  to  be  described  about  the  earth’s  axis. 

54.  The  sun’s  action  produces  a  retrograde  move¬ 
ment  of  the  equinoctial  points,  which  is  nearly,  but 
not  quite  uniform.  This  movement  may  be  separated 
into  two  parts;  one  a  continued  mean  precession  of  the 
equinoxes;  the  other  an  inequality  in  the  precession 
called  the  Solar  Nutation  in  precession.  The  ine¬ 
quality  in  the  sun’s  action  occasions  a  very  small 
change  in  the  obliquity  of  the  ecliptic,  called  the  Solar 
N utation  in  the  obliquity. 

55.  The  moon’s  action  produces  effects  similar  to 
those  produced  by  the  sun,  only  greater  in  degree. 
One  effect  is  a  mean  precession  of  the  equinoxes, 
which  combined  with  the  mean  precession  produced  by 
the  sun,  forms  the  whole  Mean  Precession  of  the 
Equinoxes.  Another  effect  is  an  inequality  in  the 
precession  called  the  Lunar  JV* utation ,  and  sometimes, 
the  Equation  of  the  Equinoxes. 

SECULAR  VARIATION  OF  THE  OBLIQUITY  OF  THE 
ECLIPTIC. 

56.  The  orbits  of  the  planets  not  coinciding  with 
the  plane  of  the  ecliptic,  their  actions  on  the  earth  tend 
to  make  it  quit  that  plane.  The  effect  is,  a  small  va¬ 
riation  in  the  position  of  the  plane  of  the  ecliptic. 
From  these  causes,  the  obliquity  of  the  ecliptic  has 
been,  and  still  continues  to  be,  diminished.  The  di¬ 
minution  at  the  present  period  is  about  52"  in  a  cen¬ 
tury.  In  process  of  time  the  same  causes  must  pro¬ 
duce  an  increase  in  the  obliquity. 

57.  The  secular  variation  of  the  obliquity  of  the 


23  6 


ASTRONOMY. 


ecliptic  was  less  in  former  ages  than  it  is  at  present. 
It  has  now  acquired  nearly  its  greatest  value,  and  will 
begin  to  decrease  about  the  23d  century  of  our  era. 
Lagrange  has  shown  that  the  total  diminution  in  the 
obliquity,  reckoning  from  that  in  1700,  must  be  less 
than  5°  23'. 


DIURNAL  ROTATION. 

58.  It  is  proved  by  minute  investigation  that  the  ac- 
>  tions  of  the  sun  and  moon,  combined  with  the  change 

in  the  position  of  the  ecliptic,  must  produce  changes 
in  the  duration  of  a  revolution  of  the  earth  on  its  axis, 
that  is,  in  the  length  of  the  day.  But  the  same  inves¬ 
tigation  also  proves  that  those  changes  are  so  indefi¬ 
nitely  minute,  that,  being  periodical,  they  can  never 
become  sensible,  even  to  the  nicest  observation. 

59.  When,  from  the  washing  of  rains,  or  from  other 
causes,  any  matter  is  made  to  descend  from  the  higher 
parts  of  mountains  to  a  position  that  is  nearer  the 
earth’s  axis,  its  velocity  will  be  diminished,  and  the 
velocity  lost,  being  communicated  to  the  mass,  must 
tend  to  accelerate  the  diurnal  motion.  But  no  changes 
known  to  us,  in  the  position  of  the  matter  of  the  earth, 
can  ever  produce  any  sensible  alteration  in  the  earth’s 
rotation  on  its  axis. 

60.  The  conclusion  drawn  from  a  full  examination 
of  the  subject  is,  that  the  duration  of  the  earth’s  rota¬ 
tion  may  be  regarded  as  perfectly  unchangeable. 

-  ,  \  s 

OF  THE  TIDES. 

61.  The  alternate  rise  and  fall  of  the  surface  of  the 
ocean,  twice  in  the  course  of  a  lunar  day,  or  of  24  h. 


CHAPTER  XVII.  237 

50  in.  48.  sec.  of  mean  solar  time,  is  the  phenomenon 
known  by  the  name  of  the  Tides . 

62.  The  time  from  one  high  water  to  the  next,  is, 
at  a  mean,  12  h.  25  m.  24  sec.  The  instant  of  low 
water  is  nearly,  but  not  exactly,  in  the  middle  of  this 
interval;  the  tide,  in  general,  taking  nine  or  ten  mi¬ 
nutes  more  in  ebbing  than  in  flowing. 

63.  The  time  of  high  w  ater  is  principally  regulated 
by  the  position  of  the  moon,  and  in  general,  in  the 
open  sea,  is  from  two  to  three  hours  after  that  body 
has  passed  the  meridian,  either  above  or  below  the 
horizon.  But  on  the  shores  of  the  larger  continents, 
and  where  there  are  shallows  and  obstructions  to  the 
motion  of  the  water,  the  interval  between  the  time  of 
the  moon’s  passage  of  the  meridian,  and  the  time  of 
high  water,  is  very  different  at  different  places.  The 
difference  is  so  great,  that  at  many  places  the  time  of 
high  water  seems  to  precede  the  moon’s  passage. 

Fol-  any  given  place,  the  time  of  high  water  is  al¬ 
ways  nearly  at  the  same  distance  from  that  of  the 
moon’s  passage  over  the  meridian. 

64.  Though  the  tides  seem  to  be  chiefly  regulated 
by  the  moon,  they  appear  also,  in  some  degree  to  be 
under  the  influence  of  the  sun.  Thus,  at  the  syzigies, 
when  the  sun  and  moon  are  on  the  meridian  together, 
supposing  other  circumstances  to  be  the  same,  the  tides 
are  the  highest;  at  the  quadratures,  when  the  sun  and 
moon  are  90°  distant,  the  tides  are  the  least. 

65.  The  tides  about  the  time  of  the  syzigies  are 
called  the  Spring  Tides ;  and  those  about  the  time  of 
the  quadratures,  are  called  the  JYeap  Tides. 

66.  The  highest  of  the  spring  tides  or  the  lowest  of 
the  neap  tides,  is  not  the  tide  that  has  place  nearest 


ASTRONOMY'. 


338 

the  syzigy  or  quadrature,  but  is  in  general  the  third, 
and  in  some  cases,  the  fourth  following  tide. 

At  Brest,  in  France,  the  tides  of  the  syzigies  rise  to 
the  height  of  19.317  feet,  and  those  of  the  quadratures 
only  to  9-1^1  feet;  which  is  not  quite  half  the  former 
quantity.  In  the  Pacific  Ocean,  the  rise  in  the  first 
case  is  5  feet,  and  in  the  second,  between  2  and  2.5 
feet. 

67.  The  height  of  the  tide  changes  with  a  change  in 
the  moon’s  distance  from  the  earth.  Other  circum¬ 
stances  being  the  same,  the  tide  is  highest  when 
the  moon  is  in  perigee,  and  the  least  when  she  is  in 
apogee. 

The  tides  also  depend  on  the  sun’s  distance  from 
the  earth  but  in  a  less  degree,  than  on  that  of  the  moon. 
In  our  winter  the  spring  tides  are  greater  than  in  the 
summer,  and  the  neap  tides  smaller. 

68.  The  tides  depend,  to  some  extent,  on  the  posi¬ 
tions  of  the  sun  and  moon  with  respect  to  the  equator. 
When  the  moon  is  in  the  northern  signs,  the  tide  of 
the  day,  in  all  the  northern  latitudes,  is  somewdiat 
greater  than  that  of  the  night.  The  contrary  has  place 
when  the  moon  is  in  the  southern  signs. 

69.  If  the  tides  be  considered  relatively  to  the  whole 
earth,  and  to  the  open  sea,  there  is  a  meridian,  about 
30°  eastward  of  the  moon,  where  it  is  always  high 
water,  both  in  the  hemisphere  w  here  the  moon  is,  and 
in  the  opposite  one.  On  the  wrest  side  of  this  meri¬ 
dian  the  tide  is  flowing,  on  the  east,  it  is  ebbing;  and 
on  the  meridian  at  right  angles  to  the  same,  it  is  low 
water.  In  consequence  of  the  earth’s  diurnal  rota¬ 
tion,  these  meridians  move  westward;  but  they  pre¬ 
serve  nearly  the  same  distance  from  the  moon,  only  ap- 


CHAPTER  XVII. 


289 

proaching  a  little  nearer  to  her  at  the  syzigies,  and 
going  farther  off  at  the  quadratures. 

The  great  Wave  which,  in  this  manner,  constitutes 
the  tide,  is  an  undulation  in  the  waters  of  the  ocean, 
in  which  there  is  very  little  progressive  motion,  except 
when  it  passes  over  shallows,  or  approaches  the 
shores. 

70.  The  facts,  which  have  been  enumerated,  clearly 
indicate  that  the  tides  are  produced  by  the  actions  of 
the  sun  and  moon;  but  in  a  geater  degree  by  that  of 
the  moon. 

It  has  been  shown  (35)  that  the  sun’s  action,  increases  or  di¬ 
minishes  the  moon’s  gravity  to  the  earth,  according  to  her  posi¬ 
tion  with  respect  to  the  line  of  the  syzigies,  or  of  the  quadra¬ 
tures.  In  like  manner,  the  sun’s  action  increases  or  diminishes 
the  gravity  of  a  particle  of  matter  at  the  earth’s  surface,  ac* 
cording  to  its  position  with  respect  to  a  plane  passing  through  the 
centre  of  the  earth,  at  right  angles  to  the  line  joining  the  centres 
of  the  earth  and  sun.  Within  about  35  of  this  plane  on  each 
side,  the  gravity  at  the  surface  is  increased;  and  at  the  remain¬ 
ing  parts,  that  is  for  about  55°  around  the  points  in  which  the 
line  of  the  centres  intersects  the  surface,  the  gravity  is  di¬ 
minished. 

Now  as  the  particles  of  water  easily  yield  to  any  impression, 
the  surface  of  the  ocean  will,  in  consequence  of  the  change  in 
the  gravity  of  its  different  parts,  assume  a  figure  different  from 
that  which  it  would  otherwise  have.  Around  the  points  in  which 
the  line  of  the  centres  intersects  the  surface,  the  gravity  being  di¬ 
minished,  the  surface  will  be  at  a  greater  distance  from  the  cen¬ 
tre;  and  in  the  middle  parts  between  these  points,  the  gravity 
being  increased,  the  surface  will  be  nearer  the  centre.  In  con¬ 
sequence  of  the  earth’s  diurnal  rotation,  it  will  successively  be 
different  parts  of  the  surface,  that  will  thus  have  the  distance 
from  the  centre  increased  and  diminished.  From  what  has  been 
said  it  is  easy  to  perceive  that  so  far  as  it  depends  on  the  sun’s 


240 


ASTRONOMY". 


action,  it  is  high  water  at  the  same  time  in  opposite  parts  of  the 
earth;  and  that  the  consecutive  high  waters  must  follow  each 
other  at  intervals  of  half  a  solar  day. 

The  moon  produces  effects  exactly  similar  to  those  of  the  sun, 
but  much  greater  in  degree,  and  succeeding  one  another  at  in¬ 
tervals  of  half  a  lunar  day. 

71.  At  the  time  of  the  syzigies  the  actions  of  the 
snn  and  moon  are  combined  in  producing  the  tides; 
but  at  the  quadratures  they  act  in  opposition  to  each 
other.  The  result  is,  much  greater  tides  at  the  syzigies 
than  at  the  quadratures.  Observations  have  made 
known  that  the  former  are  to  the  latter,  nearly  as  2  to 
1.  Consequently  the  effect  of  the  moon’s  action  must 
be  to  that  of  the  sun,  nearly  as  3  to  1. 

72.  The  relative  effects  of  the  actions  of  the  sun  and 
moon  in  producing  the  tides,  must  depend  on  their  dis¬ 
tances  and  masses;  and  as  their  distances  and  relative 
effects  are  known,  their  relative  masses  may  from 
thence  be  determined. 

73.  Great  extent  is  necessary,  in  order  that  the  sea 
should  be  sensibly  affected  by  the  actions  of  the  sun 
and  moon;  for  it  is  only  by  the  inequality  of  that 
action,  on  different  parts  of  the  mass  of  waters,  that 
their  equilibrium  is  disturbed;  and  this  inequality  can 
not  sensibly  have  place,  unless  a  great  extent  of  water 
be  included. 

74.  The  tides  which  are  experienced  in  narrow  seas, 
and  on  shores  far  from  the  main  body  of  the  ocean, 
are  not  produced  in  those  seas  by  the  direct  actions  of 
the  sun  and  moon,  but  are  waves  propagated  by  the 
great  diurnal  undulation. 


»JI  Oftf  14 
i  tut  a  r«M  ••  •  ir 

APPENDIX 


TO 

PART  I. 

Containing  Trigonometrical  Formulce;  and  Two 
Propositions  in  Conic  Sections . 

Many  of  the  Trigonometrical  Formulas  included  in  the  follow¬ 
ing  collection,  are  used  in  the  present  work.  They  are  intro¬ 
duced  here  and  numbered  in  order  to  facilitate  the  references. 
Their  demonstrations  may  be  seen  in  any  complete  treatise  on 
Trigonometry.  Nearly  all  of  them  are  contained  and  demon¬ 
strated  in  a  good  work  on  the  subject  by  Lacroix,  which  has 
been  translated  and  published  at  Cambridge,  New  England. 
Those  which  are  not  contained  in  that  work,  are  easily  de¬ 
duced  from  others  that  are. 


For  a  single  arc  or  angle  a,  the  radius  being  =  1 . 


1. 

sin 

2  a 

.  COS  2 

a  =  1 

7. 

sin 

a  r= 

2  sin 

h  a  cos  £  a 

2 

sin 

a  = 

=  tan  a  cos  a 

8. 

cos 

a  ~ 

1  — 

2  sin  2  i  a 

tan  a 

9. 

cos 

a  = 

2  cos 

\ 2  h  a —  1 

3. 

sin 

a 

=  - 

v'  1 

tan  2  a 

10. 

tan 

1  a  . 

-- 

sin  a 

1 

1  +  cos  a 

4. 

cos 

a  : 

sin  a 

v  1 

4-  tan  2  a 

11. 

cot 

2  a  = 

1  - 

—  cos  a 

5. 

tan 

a 

sin  a 

12. 

1 

—  cos  a 

cos  a 

tan 

a 

1 

4-  cos  a 

6. 

cot 

a 

1 

cos  a 

tan  a  sin  a 

For  two  arcs  a  and  b ,  of  which  a  is  supposed  to  be  the  greater. 

32 


212 


APPENDIX  TO  PART  I. 


13.  sin  (a  ±  b)  =  sin  a  cos  b  ±  cos  a  sin  b 

14.  cos  (a  ±  b)  =  cos  a  cos  b  sin  a  sin  b 

15.  tan  («±  6)  =  .  tan  a  *  tan  b 

1  -f  tan  a  tan  b 

16.  sin  a  cos  b  —  ^  sin  (a  4-  b)  4.  \  sin  (a  —  b) 

IT.  cos  a  sin  6  =  §  sin  ( a  4.  6)  —  £  sin  (a  —  6) 

18.  sin  a  sin  6  =  §  cos  (a —  b)  —  5  cos  ;  a  4-  6) 

19.  cos  a  cos  b  —  h  cos  (a  —  b)  +  \  cos  (a  4.  6) 

20.  sin  a  4-  sin  b  =  2  sin  £  (a  4-  6)  cos  §  (a  —  b) 

21.  stn  a —  sin  6  =  2  cos  \  (a  4-  6)  sin  h  (a  —  b) 

22.  cos  b  4-  cos  a  =  2  cos  h  (a  4  6)  cos  \  (a  —  b) 

23.  cos  b  —  cos  a  =  2  sin  §  (a  4-  b)  sin  \  (a  —  b) 

24.  tau  a  -f  tan  I 


25.  tan  a  —  tan  b  — 

26.  cot  b  4-  cot  a  = 

27.  cot  b  —  cot  a  = 


29.  tan  §  (a —  6)  = 

30.  cot  5  («  +  ^)  = 

31.  cot  §  (a  —  5)  = 


32. 


tan  3  fa  +  b) 


sin  (a  4 

b) 

cos  a  cos  b 

sin  (a  — 

- b ) 

cos  a  cos  b 

sin  (a  4 

- b ) 

sin  a  sin  b 

sin  (a  — 

■5) 

sin  a  sin  b 

sin  a  4. 

sin  b 

cos  a  + 

cos  b 

sin  a  — 

sin  b 

cos  a  4- 

cos  b 

sin  a  — 

sin  b 

cos  b  — 

cos  a 

sin  a  4 

sin  b 

cos  b  — 

cos  a 

sin  a 

4  sin 

sm  a 


—  sin  b 


tan  h  (a  —  b ) 

For  a  Spherical  Triangle,  in  which  A,  B,  and  C  are  the  ai 
gles,  and  a,  b ,  and  c,  the  opposite  sides,  as  in  Fig.  53. 

33.  sin  A  sin  b  =  sin  B  sin  a 

34.  cos  a  =  cos  A  sin  b  sin  c  4-  cos  b  cos  c 

35.  cos  A  =  cos  a  sin  B  sin  C  —  cos  B  cos  C 


APPENDIX  TO  PART  I, 


243 


36.  cot  a  — 

37.  cot  A  = 


cot  A  sin  B  -f-  cos  B  cos  c 
sin  c 

cot  a  sin  &  —  cos  C  cos  b 


38.  sin  i  A  =  y/ 


sin  C 

sin  \  ^  a  -|-  b  —  c)  sin  h  (a  c  bf 

sin  b  sin  c 

C0S  i  (B  —  A) 


40.  tan  \  (b  —  a)  = 

41.  tan  §  (B  4-  A) 


cos 

i  (B  +  A) 

sin 

£ 

i  (B  —  A) 

sin 

2  (B  -f  A) 

h  C 

COS  2  (b  — 

Z  ^ 

cos  h  (b 

r  sin  2  ( b  —  a) 

a) 


43. 


44. 


cot  I  C  =  tan  h  (B 


sin  h  (b  a ) 

As  sin  j  (6  +  a) 
sin  4  (6  • —  a) 


cot  ^  C  =  tan  i  (B  +  A)  C0SJJ1  +tt) 

cos  5  (b  —  a) 

tan  he  —  tan  h  (b  —  a)  .g. 

J  sin  h  (B  —  A) 

tan  h  c  =  tan  h  (b  +  a)  C0S- 

cos  2  (B  —  A) 


For  a  right  angled  spherical  triangle  in  which  C  is  the  right 
angle,  and  the  opposite  side  c,  the  hypothenuse,  as  in  Fig.  54. 

45.  cos  c  =  cos  a  cos  b  48.  tan  a  =  sin  b  tan  A 

46.  cos  c  =  cot  A  cot  B  49.  tan  a  =  cos  B  tan  c 

47.  sin  a  =  sin  c  sin  A  50.  cos  A  =  sin  B  cos  a 

51.  If  ADBL  Fig.  55,  be  an  Ellipse,  AB4/ie  transverse  axis , 
E  and  F  the  foci,  C  the  centre,  and  D  a  point  in  the  curve,  then , 

AC2  —  EC2 


ED  = 


AC  —  EC  cos  AED* 


Let  DH  be  perpendicular  to  AB.  Then, 

ED2  =  DH2  +  EH2  =  DH2  +  (EC  +  CH)2 


APPENDIX  TO  PART  I. 


244 


=  DH2  +  EC2  +  2  EC  x  CH  +  CH% 
and  FD2  =  DH2  +  FH2  =,  DH2  +  (EC  — CH)2 
DH2  +  EC2  —  2EC  x  CH  +  CH2. 
Hence  by  subtraction, 

ED2  — FD2  4EC  x  CH. 

But  ED2— FD2=  (ED  +  FD)  x  (ED  —  FD) 
=  2AC  x  (ED  —  FD). 

Therefore, 

2AC  x  (ED  —  FD)  =  4  EC  x  CH, 

Or,  ED  —  FD  =  2—  CH. 

AC 


But  ED  f  FD  =  2AC. 
Hence  by  addition, 

2  ED  =  2AC  + 


2EC  x  CH 
AC 


ED 


AC 


EC  x  CH 
AC 


(A) 


ED  x  AC  =  AC2  +  EC  x  CH 

=  AC2  +  EC  x  (EH  — EC) 

=  AC2  +  EC  x  (ED  cos  AED  —EC) 

=  AC2  +  EC  x  ED  cos  AED  —  EC2, 
ED  x  AC  —  ED  x  EC  cos  AED  =  AC2  —  EC2 
ED  x  (AC  —  EC  cos  AED)  =  AC2  —  EC2 

ED  =  AC2  —  EC2 

AC  —  EC  cos  AED 

52.  If  the  circle  AGBM,  Fig.  55,  be  described  about  AB,  as 
diameter y  and  HG  be  produced  to  meet  it  in  G,  then , 

ED  AC  — EC  cos  BCG. 

From  the  preceding  demonstration,  we  have  (A), 


ED  =  AC  +  _LS  9.1* 

AC 


=  AC  + 
=  AC  + 


EC  x  CG  cos  ACG 
AC 

EC  x  AC  cos  ACG 


AC 

=  AC  +  EC  cos  ACG 
=  AC  — EC  cos  BCG. 


AN 


ELEMENTARY  TREATISE 


i'"'\  .  rVA&  ,  .  i 

ASTRONOMY.  . 

PART  II. 

CATALOGUE  of  the  Tables  with  observations  respecting 
.some  of  them. 


TABLE  I. 

Latitudes,  and  Longitudes  from  the  Meridian  of  Greenwich,  of 
some  Cities  and  other  conspicuous  places. 

TABLE  II. 

Mean  Astronomical  Refractions. 

TABLE  III. 

Mean  Right  Ascensions  and  Declinations  of  some  of  the  Fixed 
Stars,  for  the  beginning  of  1820,  with  their  Annual  Variations. 

TABLE  IV. 

Mean  New  Moons  &c.  in  January.  The  time  of  mean  new 
moon  in  January  of  each  year  has  been  diminished  by  15  hours, 
which  has  been  added  to  the  equations  in  Table  VII.  Thus, 
4  h.  20  m.  has  been  added  to  the  first  equations;  10  h.  10  m.  to 
the  second;  10  minutes  to  the  third;  and  20  minutes  to  the 
fourth.  By  this  means  the  equations  for  finding  the  approximate 
time  of  new  or  full  moon,  are  all  made  additive. 


246 


ASTRONOMY. 


TABLES  V,  VI,  and  VII. 

These  tables  are  used  with  the  preceding  one,  in  finding,  nearly, 
the  true  time  of  new  or  full  moon. 

TABLE  VIH. 

Mean  Longitudes  and  Latitudes  of  some  of  the  Fixed  Stars, 
for  the  beginning  of  1810,  with  their  Annual  Variations. 

TABLE  IX. 

Sun’s  Mean  Longitude,  the  Longitude  of  the  Perigee,  and 
Arguments  for  finding  some  of  the  small  equations  of  the  sun’s 
place.  They  are  all  given  for  mean  noon  at  the  meridian  of 
Greenwich,  on  the  first  of  January  for  common  years,  and  on  the 
second  of  January  for  bissextiles.  The  sun’s  longitudes  and  the 
longitudes  of  his  perigee  have,  each,  been  diminished  by  2°.  As 
each  is  diminished  by  the  same  quantity,  the  mean  anomaly, 
which  is  obtained  by  subtracting  the  longitude  of  the  perigee, 
from  the  sun’s  longitude,  and  which  is  the  argument  for  the  equa¬ 
tion  of  the  centre,  is  not  affected.  The  Argument  I,  is  for  the 
equation  depending  on  the  action  of  the  moon;  Argument  II,  is 
for  that  depending  on  the  action  of  Jupiter;  Argument  III,  is  for 
that  depending  on  the  action  of  Venus;  and  Argument  N,  is  for 
the  Nutation,  or  equation  of  the  equinoxes. 

Of  the  2°  which  has  been  subtracted  from  the  sun’s  mean  lon¬ 
gitudes,  1°  59'  30"  is  added  to  the  equation  of  the  centre,  and  10" 
to  each  of  the  small  equations  due  to  the  actions  of  the  Moon, 
Jupiter  and  Venus. 

TABLE  X. 

Motions  of  the  Sun  and  Perigee  and  change  in  the  arguments, 
for  Months. 

TABLES  XI  and  XII. 

Sun’s  Hourly  Motion  and  Semidiameter.  These  two  tables 
would,  in  order,  come  after  table  XVIII,  but  are  put  in  the 


ASTRONOMY. 


Ml 

place  which  they  occupy  with  a  view  to  convenient  arrangement 
on  the  pages. 

TABLES  XIII  and  XIV. 

Sun’s  Motions  for  Days,  Hours,  Minutes  and  Seconds. 
TABLE  XV. 

Equation  of  the  Sun’s  Centre. 

TABLE  XVI. 

Small  equations  of  Sun’s  Longitude. 

TABLE  XVII. 

Mean  Obliquity  of  the  Ecliptic  for  the  beginning  of  each  year 
contained  in  the  table. 


TABLE  XVIII. 

Nutation  in  Longitude,  Right  Ascension  and  Obliquity  of  the 
Ecliptic. 


TABLE  XIX. 

Equation  of  Time,  to  convert  Apparent  into  Mean  Time. 
TABLE  XX. 

Epochs  of  the  Moon’s  Mean  Longitude  and  of  the  Arguments 
for  finding  the  Equations  which  are  necessary  in  determining  the 
True  Longitude  and  Latitude  of  the  Moon.  They  are  all  given 
for  mean  noon  at  the  meridian  of  Greenwich,  on  the  first  of 
January  for  common  years,  and  on  the  second  of  January,  for 
bissextiles.  The  Argument  for  the  Evection  is  diminished  by 
29',  the  Anomaly  by  1°59',  the  Argument  for  the  Variation  by 
8°  59',  the  Mean  Longitude  by  9°  44';  and  the  Supplement  of  the 
Node  is  increased  by  7'.  This  is  done  to  balance  the  quantities 
which  are  applied  to  the  different  equations  to  render  them  af¬ 
firmative. 


ASTRONOMY. 


248 

TABLES  XXI  to  XLII,  inclusive. 

These  tables  together  with  table  XX,  are  for  finding  the  Moon’s 
True  Longitude,  Latitude  and  Equatorial  Parallax. 

TABLE  XLIII. 

Reductions  of  Parallax  and  of  the  Latitude  of  a  Place.  The 
reduction  of  parallax  is  for  obtaining  the  parallax  at  any  given 
place  from  the  equatorial  parallax.  The  reduction  of  latitude  is 
for  reducing  the  true  latitude  of  a  place  as  determined  by  obser¬ 
vation,  to  the  corresponding  latitude  on  the  supposition  of  the 
earth  being  a  sphere.  The  ellipticity  to  which  the  numbers  in 
the  table  cgrresponds  is  -3^-.  This  differs  a  little  from  what  is 
believed  to  be  the  most  accurate  determinations  of  the  ellipticity, 
which  make  it  from  to  But  the  difference  is  too 

small  to  be  regarded  unless  its  value  was  known  to  a  greater  de¬ 
gree  of  precision. 

TABLES  XLIV  and  XLV. 

Moon’s  Semidiameter  and  the  augmentation  of  the  semidiame¬ 
ter  depending  on  the  altitude. 

TABLES  XLVI  to  LIV,  inclusive. 

Moon’s  Hourly  Motions  in  Longitude  and  Latitude. 

TABLE  LV. 

Contains  1 1  pages  of  the  Nautical  Almanac,  taken  from  the 
month  of  August  for  that  of  1821. 

TABLE  LVI. 

Second  differences.  This  table  is  useful  for  finding  from  the 
Nautical  Almanac,  the  Moon’s  longitude  or  latitude  for  any  time 
between  noon  and  midnight. 

TABLE  LVII. 

Logistical  Logarithms.  This  table  is  convenient  in  working 


ASTRONOMY. 


249 


proportions,  when  the  terms  are  minutes  and  seconds,  or  degrees 
and  minutes;  or  hours  and  minutes. 

TABLE  LVIII. 

Change  in  Moon’s  Right  Ascension  from  the  Sun.  This 
table  serves  to  find  the  time  of  the  moon’s  passage  over  the  meri¬ 
dian  of  any  given  place,  from  the  time  of  its  passage,  as  given  in 
the  Nautical  Almanac  for  the  meridian  of  Greenwich.  It  is  also 
convenient  in  a  calculation  for  the  rising  or  setting  of  the  moon, 
to  determine  the  correction  of  the  semi-diurnal  arc,  which  de¬ 
pends  on  the  moon’s  change  in  right  ascension  from  the  sun. 

TABLE  LIX. 

Change  in  Moon’s  Declination.  This  table  is  convenient  in 
finding  from  the  Nautical  Almanac,  the  moon’s  declination  for 
any  intermediate  time  between  noon  and  midnight. 

TABLES  LX,  to  LXIII,  inclusive. 

These  are  tables,  calculated  by  M.  Gauss,  for  finding  the  Aber¬ 
ration  and  Nutation,  of  a  Star,  in  Right  Ascension  and  Declina¬ 
tion. 


TABLE  LXIV. 

Semi-diurnal  Arcs  for  the  Latitude  of  39°  57'  North. 

SCHOLILTM.  The  tables  of  the  Sun,  which  are  those  from 
IX  to  XIX,  inclusive,  are  abridged  from  Delambre’s  Solar  Tables. 
And  those  of  the  Moon,  which  are  from  XX  to  LIV,  inclusive, 
are  abridged  from  Burckhardt’s  Lunar  Tables.  As  some  small 
equations,  and  also  the  tenths  of  seconds  are  omitted,  all  the 
quantities  obtained  from  these  tables  will  be  liable  to  small  errors. 
None  of  these  errors  will,  however,  exceed  a  few  seconds. 

It  may  be  proper  here  to  inform  the  student,  that  when,  in  the 
following  problems,  he  meets  with  the  expressions,  Sun’s  true 
longitude.  Moon’s  true  longitude,  &c.  he  is  to  understand  them  as 
implying  the  true  values  of  those  quantities  so  far  as  they  can  be 
obtained  from  the  tables  used. 

33 


250 


ASTRONOMY. 


Observations  and  Rules ,  relative  to  Quantities  with 
different  Signs. 


It  is  frequently  convenient,  in  computations,  to  designate  cer¬ 
tain  quantities  by  the  Affirmative  sign  f-,  perfixed;  and  others  by 
the  Negative  sign  —  .  Those  which  have  the  affirmative  sign 
prefixed,  are  called  Positive  or  Affirmative  quantities;  and  those 
with  the  negative  sign,  prefixed,  are  called  Negative  quantities. 

When  a  quantity  is  affirmative,  the  sign  is  frequently  omitted; 
but  when  it  is  negative,  the  sign  must  always  be  used. 

I 

To  add  quantities ,  having  regard  to  their  signs.  When  all  the 
quantities  have  the  same  sign,  add  them  as  in  common  arithmetic, 
and  prefix  that  sign  to  the  sum.  When  the  quantities  have  dif¬ 
ferent  signs,  add  the  affirmative  quantities  into  one  sum,  and  the 
negative  into  another.  Then  take  the  difference  of  these  two 
sums  and  prefix  the  sign  of  the  greater.  These  rules  will  be  il¬ 
lustrated  by  the  following  examples. 


Add  2'  11" 
7  2 

3  4 


Sum  12  17 

Add +  3'  15" 
—  8  12 
—  5  1 

+  2  17 


Add  — 3'  51" 

—  4  10 

—  1  15 


Sum — 9  16 

Add — 17'  10" 
—  4  3 

+  12  4 

+  18  59 


Add  — 7'  14" 
4-8  2 
4-  3  17 


Sum  4-4  5 

Add  4-  3'  1" 

—  1  15 
4-  4  18 

—  6  4 


Sum  —  7  11  Sum  4-  9  50  Sum  0  0 

To  Subtract  quantities ,  having  regard  to  their  signs.  Suppose 
the  sign  of  the  quantity  which  is  to  be  subtracted,  to  be  changed; 
that  is,  if  it  is  affirmative,  suppose  it  to  be  negative;  and  if  it  is 
negative  suppose  it  to  be  affirmative.  Then  proceed  as  in  the 
above  rule  for  adding  quantities.  Thus, 


ASTRONOMY. 


From  5'  10" 
Sub.  3  21 

Rem.  1  49 

From  —  8'  29" 
Sub.  —3  2 

Rem.— 5  27 


From  4'  11" 
Sub.  7  27 

Rem.  —  3  16 

From  —  2'  18" 
Sub.  —7  11 

Rem.  +  4  53 


From  -f  2'  5 " 
Sub.  —  1  11 

Rem.  -f  3  16 

From  — 4'  17" 
Sub.  +  6  21 

Rem.  —  10  3S 


To  find  the  Logarithmic  Sine ,  Cosine ,  Tangent ,  or  Cotangent 
of  an  arc ,  with  its  proper  Sign ,  from  Tables  that  extend  only  to 
each  minute  of  the  quadrant . 

JF/ien  the  given  arc  does  not  exceed  180°.  With  the  given  arc, 
or  when  it  exceeds  90°,  with  its  supplement  to  180°,  take  out 
from  the  table,  the  required,  Sine,  or  Tangent,  &c.  When  there 
are  seconds ,  take  out  the  quantity  corresponding  to  the  given  de¬ 
grees  and  minutes;  also  take  the  difference  between  this  quantity 
and  the  next  following  one,  in  the  table.  Then  60"  :  the  odd 
seconds  of  the  given  arc  :  :  the  difference  :  a  fourth  term.  This 
fourth  term,  added  to  the  quantity  taken  out,  when  it  is  in¬ 
creasing',  but  subtracted ,  when  it  is  decreasing ,  will  give  the  re¬ 
quired  quantity. 

TV  hen  the  given  arc  exceeds  180°.  Subtract  180°  from  it,  and 
proceed  as  before.  When  the  arc  exceeds  270°,  it  is  more  con¬ 
venient,  and  amounts  to  the  same,  to  subtract  it  from  360°. 

To  determine  the  Sign  of  the  quantity.  Call  the  arc  from  0°  to 
90°,  the  first  quadrant;  from  90°  to  180°,  the  second  quadrant; 
from  180°  to  270°,  the  third  quadrant;  and  from  270°  to  360°, 
the  fourth  quadrant.  Then, 

The  Sine  of  the  arc  is  affirmative  for  the  first  and  second  quad¬ 
rants;  and  negative ,  for  the  third  and  fourth. 

The  Cosine ,  is  affirmative  for  the  first  and  fourth  quadrants; 
and  negative ,  for  the  second  and  third. 

The  Tangent  and  Cotangent ,  are  affirmative  for  the  first  and 
third  quadrants;  and  negative ,  for  the  second  and  fourth. 

By  attending  to  the  preceding  rules,  the  student  will  easily 
find  the  Sine,  Cosine,  &c.  of  an  arc,  in  either  quadrant,  with  its 
appropriate  sign,  as  exemplified  in  the  following  table. 


252 


ASTRONOMY. 


Arc 

37 *  18/  21" 
114  35  10 
247  12  3G 
314  17  50 


Sine 

+  9.78252 
.  9.95872 

—  9.96470 

—  9.85475 


Cosine 

9.90060 

—  9.61916 

—  9.58811 
4-  9.84409 


Tangent 
4-  9.88193 

—  10.33956 
4-  10.37659 

—  10.01065 


Cotang. 
4-  10.1180 7 

—  9.66044 
4-  9.62341 

—  9.98935 


Note.  The  signs  are  seldom  placed  before  affirmative  loga¬ 
rithms;  but  they  must  not  be  omitted  before  negative  ones. 


The  Logarithmic  Sine ,  Cosine ,  Tangent ,  or  Cotangent  of  an 
arc  being  given ,  to  find  the  arc . 

When  the  given  quantity  can  be  found  in  the  table,  under  or 
over  its  name,  take  out  the  corresponding  arc.  When  the  given 
qu  ntitv  is  not  found  exactly  in  the  table,  and  the  arc  is  required 
to  seconds,  take  out  the  degrees  and  minutes  corresponding  to  the 
next  less  quantity,  when  that  quantity  is  increasing;  but  to  the 
next  greater  when  it  is  decreasing.  Take  the  difference  between 
the  quantity  corresponding  to  the  degrees  taken  out,  and  the  next 
following  one  in  the  table;  also  take  the  difference  between  the 
same  quantity  and  the  given  one.  Then,  the  first  difference  :  the 
second  :  :  60"  :  the  number  of  seconds  which  is  to  be  annexed  to 
the  degrees  and  minutes.  Then, 

For  a  Sine.  When  it  is  affirmative ,  the  required  arc  will  be, 
either  the  arc  found  in  the  table,  or  its  supplement  to  180°.  When 
the  sine  is  negative ,  the  required  arc  will  be,  either  the  arc  found 
in  the  table,  increased  by  180°,  or  its  supplement  to  360°. 

For  a  Cosine.  When  it  is  affirmative ,  the  required  arc  will 
be,  either  the  arc  found  in  the  table,  or  its  supplement  to  360\ 
When  the  cosine  is  negative ,  the  required  arc  will  be,  either  the 
supplement  of  the  arc  found  in  the  table,  to  180°,  or  that  arc,  in¬ 
creased  by  180°. 

For  a  Tangent  or  Cotangent  When  it  is  affirmative ,  the  re¬ 
quired  arc  will  be,  either  the  arc  found  in  the  table,  or  that  arc, 
increased  by  180°.  When  the  tangent  or  cotangent  is  negative , 
the  required  arc  will  be,  either  the  supplement  of  the  arc  found 
in  the  table,  to  180°,  or  its  supplement  to  360°. 


ASTRONOMY. 


253 

These  rules  are  exemplified  by  the  quantities  in  the  following 
table. 

Sine  +  9.78252,  arc  37°  18'  21"  or  142°  41'  39" 

Sine  —  9.85475  arc  225  42  10  or  314  17  50 

Cosine  +  9.90060  arc  37  18  18  or  322  41  42 

Cosine  —  9.61916  arc  114  35  11  or  245  24  59 

Tangent  +  9.8S1S3  arc  37  18  21  or  217  18  21 

Tangent  —1033956  arc  114  35  11  or  294  35  11 
Cotangent  +  9.62341  arc  67  12  36  or  247  12  36 

Cotangent  —  9.98935  arc  134  17  51  or  314  17  51. 

Note.  Tables  which  extend  only  to  five  decimals,  will  give 
the  arc,  for  a  tangent  or  cotangent,  true  to  the  nearest  second,  for 
a  few  degrees,  near  to  0°,  90°,  180°,  or  270  ;  for  a  sine,  near  to 
0'  or  180°;  and  for  a  cosine,  near  to  90°  or  270  .  In  other  cases 
they  can  not  be  depended  on,  to  give  the  seconds  accurately. 
They  are,  however,  sufficient  for  many  calculations;  particularly, 
when  the  nature  of  the  problem  does  not  make  it  necessary  that 
the  required  arc  or  angle  should  be  determined  with  great  ac¬ 
curacy. 

As  almost  every  mathematical  student  is  furnished  with  a  set  of 
such  tables,  and  as  an  example  worked  by  them,  will  serve  as 
well  to  illustrate  a  rule,  as  if  worked  by  those  which  are  more  ex¬ 
tensive,  they  will  be  used,  when  necessary,  in  working  the  ex¬ 
amples  and  questions  in  the  following  problems. 

Observations,  relative  to  the  Signs  of  the  Logarithms  of  Natural 
Numbers . 

When  the  logarithm  of  a  natural  number  is  used  in  calcula¬ 
tion  its  sign  is  affirmative  or  negative,  according  to  that  of  the 
number. 

When  the  natural  number  is  a  decimal,  in  order  to  avoid  a  dif¬ 
ficulty  with  respect  to  the  sign,  the  arithmetical  complement,  of 
the  index  is  used.  Thus,  when  there  is  no  cypher  between  the 
decimal  point  and  first  significant  figure,  the  index  is  9;  when 
there  is  one  cypher,  the  index  is  8;  when  there  are  two  cyphers, 
the  index  is  7;  and  so  on.  Thus, 


251* 


ASTRONOMY. 


The  logarithm  of  .27  is  9.43136 

of  .027  is  8.43136 

of— .027  is— 8.43136 
of  .0027  is  7.43136 

of —  .0027  is  — 7.43136 

When,  in  order  to  get  the  product  or  quotient  of  quantities 
several  logarithms,  or  logarithms  and  the  arithmetical  comple¬ 
ments  of  logarithms  are  added  together,  if  they  are  all  affirmative, 
or  if  there  is  an  even  number  of  negative  ones,  the  resulting  lo¬ 
garithm  will  be  affirmative;  but  if  there  is  an  odd  number  of  ne¬ 
gative  ones,  the  resulting  logarithm  will  be  negative. 

When  the  resulting  logarithm  of  a  calculation,  is  the  logarithm 
of  a  natural  number,  the  number  will  be  affirmative  or  negative, 
according  to  the  sign  of  the  logarithm. 

When  in  any  of  the  calculations  on  the  following  problems,  the 
resulting  logarithm  is  the  logarithm  of  a  natural  number,  if  the 
index  is  9,  or  near  to  9,  as  8,  7,  &c.  the  number  will  be  a  decimal. 
When  the  index  is  9,  there  must  be  no  cypher  between  the 
decimal  point  and  first  significant  figure.  When  the  index  is  8, 
there  must  be  one  cypher;  when  the  index  is  7,  there  must  be  two 
cyphers;  and  so  on. 


PROBLEMS  FOR  MAKING  VARIOUS  AS¬ 
TRONOMICAL  CALCULATIONS. 

PROBLEM  I* 

To  work ,  by  logistical  logarithms ,  a  proportion ,  the  terms  of 
which  are  minutes  and  seconds  of  a  degree  or  of  time ,  or  hours 
and  minutes. 

With  the  minutes  at  the  top  and  seconds  at  the  side,  or  if  a 
term  consists  of  hours  and  minutes,  with  the  hours  at  the  top  and 

*  Perhaps  in  strict  language,  this  and  a  few  of  the  following  problems  are 
not  properly  called  Astronomical.  They  are  however  for  performing  sub¬ 
sidiary  operations,  in  astronomical  calculations. 


ASTRONOMY* 


255 


minutes  at  the  side,  take  from  table  LVII  the  logistical  logarithms 
of  the  three  given  terms,  and  proceed  in  the  usual  manner  of 
working  a  proportion  by  logarithms.  The  quantity,  in  the  table, 
corresponding  to  the  resulting  logarithm  will  be  the  fourth  term. 
Note  1.  The  logistical  logarithm  of  60'  is  0. 

2.  The  student  will  easily  perceive  that  proportions  that 
are  worked  by  logistical  logarithms,  may  also  be  worked  by 
the  common  rule  in  arithmetic. 

Exam.  1.  When  the  moon’s  hourly  motion  is  31'  57",  what 
is  its  motion  in  39  m.  22  sec.?  Ans.  20'  58". 

As  60  m.  -  -  -  0 


:  39  m.  22  sec  - 

1830 

:  31'  57"  - 

-  2737 

:  20'  58" 

4567 

2.  If  the  moon’s  declination  change  2°  29'  in  12  hours,  what 
will  be  the  change  in  8  h.  21m.  Ans.  V  44'. 

As  12  h.  -  -  -  6990 

:  8h.  21m.  -  -  8565 

: :  2°  29'  -  -  13831 


22396 


:  1°  44'  -  -  15406 

3.  When  the  sun’s  hourly  motion  is  2'  31",  what  is  its  motion 
in  17  m.  18  sec.  Am.  0'  44" 

4.  When  the  sun’s  declination  changes  22'  14"  in  24  hours, 
what  is  its  change  in  19  h.  25  m.?  Ans .  17'  59" 

PROBLEM  II. 

From  a  table  in  which  quantities  are  given^for  each  Sign  and 
Degree  of  the  circle ,  to  find  the  quantity  corresponding  to  Signs , 
Degrees ,  Minutes  and  Seconds. 

Take  out,  from  the  table,  the  quantity  corresponding  to  the 
given  signs  and  degrees;  also  take  the  difference  between  this 


256 


ASTRONOMY. 


quantity  and  the  next  following  one.  Then,  60'  :  odd  minutes 
and  seconds  :  :  this  difference  :  a  fourth  term.  This  fourth  term 
added  to  the  quantity  taken  out,  when  the  quantities  in  the  table 
are  increasing;  but  subtracted,  when  they  are  decreasing,  will 
give  the  required  quantity. 

Note  1.  When  the  quantities  change  but  little  from  degree  to 
degree,  the  required  quantity  may  frequently  be  estimated,  with¬ 
out  the  trouble  of  making  a  proportion. 

Note  2.  The  given  quantity  with  which  a  quantity  is  taken 
from  a  table,  is  called  the  Argument. 

Note  3.  In  many  tables,  the  argument  is  given  in  parts  of  the 
circle,  supposed  to  be  divided  into  a  100,  a  1000,  or  10000,  &c. 
parts.  The  method  of  taking  quantities  from  such  tables  is  the 
same  as  is  given  in  the  above  rule;  except  that  when  the  argu¬ 
ment  changes  by  10,  the  first  term  of  the  proportion  must  be  10, 
and  the  second,  the  odd  units;  when  the  argument  changes  by 
100,  the  first  term  must  be  100,  and  the  second,  the  odd  parts 
between  hundreds;  and  so  on 

Exam.  1.  Given  the  argument  Is  9°  31'  26",  to  find  the  corres¬ 
ponding  quantity  in  table  XXXII.  Ans.  11°  13'  17". 

Is  9°  gives  11°  IT  15". 

The  difference  between  IT  IT  15"  and  the  next  following 
quantity  in  the  table  is  5'  9". 

As  60'  :  3T  26"  :  :  5'  9"  :  2' 42".* 

To  11°  IT  15" 

Add  2  42 


11  13  57 

2.  Given  the  argument  10s  13°  16'  54",  to  find  the  corres¬ 
ponding  quantity  in  table  XXXV.  Ans.  93°  32'  37". 

10s  13°  gives  93'  33'  40". 

The  difference  between  93°  33'  40"  and  the  next  following 
quantity  in  the  table,  is  3'  43". 

•  The  student  can  work  the  proportion,  either  by  common  arithmetic,  or 
by  logistical  logarithms,  as  he  may  prefer. 


ASTRONOMY. 


257 


As  60'  :  16' 

54" 

:  :  3' 

43" 

From 

93° 

33' 

40' 

Take 

1 

3 

93 

32 

37 

3.  Given  the  argument  4s  11°  57'  10",  to  find  the  corres¬ 
ponding  quantity  in  table  XV.  Jins.  3°  24'  12". 

4.  Given  the  argument  3721,  to  find  the  corresponding  quantity 
in  table  XXV.  Ans.  4'  52" 

PROBLEM  III. 

To  convert  Degrees ,  Minutes  and  Seconds  of  the  Equator  into 
Time. 

Multiply  the  quantiyby  4,  and  call  the  product  of  the  seconds, 
thirds;  of  the  minutes,  seconds;  and  of  the  degrees,  minutes. 

Exam.  1.  Convert  72°  17'  42",  into  time. 

72°  17'  42" 

4 

4  h,  49  m.  10  sec.  48'".  =  4  h.  49  m.  1 1  sec.  nearly. 

2.  Convert  117°  12' 30",  into  time.  Jins .  7h.  48  m,  50  sec. 

3.  Convert  21°  52'  27",  into  time.  Jins.  1  h.  27  m.  30  see. 

PROBLEM  IV. 

To  convert  Time ,  into  Degrees ,  Minutes  ond  Seconds . 

Reduce  the  time  to  minutes,  or  minutes  and  seconds;  divide  by 
4,  and  call  the  quotient  of  the  minutes,  degrees;  of  the  seconds, 
minutes;  and  multiply  the  remainder  by  15,  for  the  seconds. 

Exam.  1.  Convert  5h.  41  m.  10  sec.  into  Degrees,  &c. 

h.  m.  sec. 

5  41  10 

60 

4)341  10 


34 


85°  17'  30" 


258 


ASTRONOMY. 


2 .  Convert  7h.  48  m.  50  sec.  into  Degrees,  &c.  Ans. 
117°  12'  SO  '. 

3.  Convert  11  h.  17  m.  21  sec.  into  Degrees,  &c.  Ans. 
169°  20'  15". 

PROBLEM  V. 

The  Longitude  of  two  Places ,  and  the  Time  at  one  of  them  being 
given ,  to  find  the  corresponding  Time  at  the  other. 

Express  the  given  time  astronomically.  Thus,  when  it  is  in 
the  morning,  add  12  hours,  and  diminish  the  number  of  the  day, 
by  a  unit.  When  the  given  time  is  in  the  afternoon,  it  is  already, 
in  astronomical  time. 

Find  the  difference  of  longitude  of  the  two  places,  by  sub¬ 
tracting  the  less  longitude  from  the  greater,  when  they  are  both 
of  the  same  name,  that  is  both  east,  or  both  west;  but  by  adding 
the  two  longitudes  together,  when  they  are  of  different  names. 
When  one  of  the  places  is  Greenwich,  the  longitude  of  the  other, 
is  the  difference  of  longitude. 

Then,  if  the  place,  at  which  the  time  is  required,  is  to  the  east 
of  the  other  place,  add  the  difference  of  longitude,  in  time,  to  the 
given  time;  but  if  it  is  to  the  i vest,  subtract  the  difference  of  lon¬ 
gitude,  from  the  given  time.  The  sum  or  remainder  is  the  re¬ 
quired  time. 

Note.  The  longitudes  of  the  places  mentioned  in  the  following 
examples,  are  given  in  table  1. 

Exam.  1.  When  it  is  August  8th,  2  h.  12  m.  17  sec.  A.  M. 
at  Greenwich,  what  is  the  time,  as  reckoned  at  Philadelphia? 

d.  h.  m.  sec. 

Time  at  Grenwich,  August,  7  14  12  17 

Diff.  of  Long.  -  -  .  5  0  46 

Time  at  Philadelphia,  -  7  91131P.M. 

2.  When  it  is  April  11th,  3  h.  15  m.  20  sec.  P.  M.  at  New 
York,  what  is  the  corresponding  time  at  Greenwich? 


ASTRONOMY. 


259 


d.  h.  m.  sec. 

Time  at  New  York,  April,  11  3  15  20 
Differ,  of  Long.  -  -  4  56  4 

Time  at  Greenwich,  11  8  1 1  24  P.  M. 


3.  When  it  is  Sept.  10th.  3h. 
what  is  the  time  as  reckoned  at  ] 

Longitude  of  Paris, 

do.  of  New-Haven, 

Diff.  of  Long. 

Time  at  Paris,  September, 
Diff.  of  Long. 

Time  at  New-Haven, 

Or  September  10th,  10 


20  m.  35  sec.  P.  M.  at  Paris, 
iw-Haven? 
h.  m.  sec. 

0  9  21  E. 

4  51  52  W. 


5  1  13 

d.  h.  m.  sec. 

10  3  20  35 

5  1  13 

9  22  19  22 
i.  19  m.  22  sec.  A.  M. 


4.  What  it  is  January  15th,  9  h.  12  m.  10  sec.  P.  M.  at 
Washington,  what  is  the  corresponding  time  at  Berlin?  Ans.  Sept. 
16th,  3  h.  13  m.  21  sec.  A.  M. 

5.  When  it  is  Oct.  5th.  7  h.  8  m.  A.  M.  at  Quebec,  what  is 
the  time  at  Richmond?  Ans.  Oct.  5th.  6  h.  40  m.  47  sec.  A.M. 

6.  When  it  is  noon,  of  the  10th  of  June  at  Greenwich,  what 
is  the  time  at  Philadelphia?  Ans.  June  10th,  6h.  59  m.  14  sec. 
A.  M. 


PROBLEM  VI. 

The  Apparent  Time  being  given ,  to  find  the  corresponding 
Mean  Time ;  or  the  Mean  Time  being  given,  to  find  the  Ap¬ 
parent. 

When  the  given  time  is  not  for  the  meridian  of  Greenwich,  re¬ 
duce  it  to  that  meridian  by  the  last  problem.  Then,  from  the 
tables  take  out  the  sun’s  Mean  Longitude  corresponding  to  this 
time.  Thus,  from  table  IX,  take  the  longitude,  corresponding  to 
the  given  year;  and  from  tables  X,  XIII,  and  XIY,  take  the 


260  ASTRONOMY. 

motions  in  longitude,  for  the  months,  days,  &c.  The  sum,  re¬ 
jecting  12  signs,  when  it  exceeds  that  quantity,  will  be  the  Sun’s 
Mean  Longitude  as  given  by  the  tables. 

With  the  Sun’s  Mean  Longitude,  thus  found,  take  the  Equation 
of  Time  from  table  XIX.  Then,  when  Apparent  Time  is  given, 
apply  the  equation  with  the  Sign  it  has  in  the  table;  but  when 
Mean  Time  is  given,  apply  it  with  a  contrary  Sign;  the  result 
will  be  the  Mean  or  Apparent  Time,  required. 

Note  1.  In  taking  the  sun’s  longitude  from  the  tables,  it  is  not 
necessary  to  regard  the  seconds  in  the  given  time. 

Note  2  The  Sun’s  mean  longitude,  found  from  the  tables  in 
this  work,  is  always  two  degrees  less  than  its  true  value;'butthis 
difference  is  allowed  for,  in  arranging  the  numbers  in  table  XIX. 

Note  3.  The  Equation  of  Time  is  given  in  the  Nautical 
Almanac  for  each  day  in  the  year,  at  apparent  noon,  on  the  me¬ 
ridian  of  Greenwich,  and  can  easily  be  found  for  any  intermediate 
time  by  proportion.  When  Apparent  Time  is  given  to  find 
Mean,  the  equation  is  to  be  applied  according  to  its  Title;  but 
W'hen  Mean,  is  given,  to  find  Apparent,  it  must  be  applied,  con¬ 
trary  to  its  Title.  The  Equation  is  given  on  the  second  page  of 
each  month.  See  the  second  page  of  table  LV. 

Exam.  1.  On  the  15th  of  August,  1821,  when  it  isSh. 
15  m.  12  sec.  A.  M.  mean  time  at  Philadelphia,  what  is  the  ap¬ 
parent  time  at  the  same  place? 

d.  h.  m.  sec. 

Time  at  Philadelphia,  August  1821,  14  20  15  12 

Diff.  of  Long.  -  -  -  -  5  0  46 

Time  at  Greenwich  -  -  -15115  58 

M.  Long. 


1821 

i 

CD 

GO 

48' 

19' 

August 

6  28 

57 

26 

15d. 

13 

47 

57 

lh.  - 

2 

28 

16  m. 

. 

39 

Mean  Long. 


4  21  36  49 


ASTRONOMY.  261 

The  equation  of  time  in  table  XIX,  corresponding  to 
4s  21°  36'  49''  is  +  4  m.  13 sec. 

d.  h.  m.  sec. 

Mean  Time  at  Philadelphia,  August  1821, 14  20  15  12 
Equation  of  time,  sign  changed  -  —  4  13 

Apparant  time  -  -  -  -  14  20  10  59 

Or,  August  1821,  15  d.  8  h.  10  m.  59  sec.  A.  M. 

2.  On  the  18th  of  October,  1821,  when  it  is  3h.  21m. 
17  see.  P.  M.  apparent  time,  at  Philadelphia,  what  is  the  mean 
time  at  Greenwich? 

d.  h.  m.  sec. 

Time  at  Philadelphia,  October  1821,  18  3  21  17 
Diff.  of  Long.  -  -  5  0  46 

Time  at  Greenwich  -  -  18  8  22  3 

M.  Long. 

1821  -  9s  8°  48' 19" 

October  8  29  4  54 

18  d.  -  -  16  45  22 

8  h.  19  43 

22  m.  -  54 

M.  Long.  6  24  59  12Equat.oftime— 14m. 48 sec. 

d.  h.  m.  sec. 

Appar.  Time  at  Greenwich,  Oct.  1821,  18  8  22  3 
Equation  of  time  -  -  -  —  14  48 

Mean  Time  at  Greenwich  -  -  188  715 

3.  On  the  15th  of  May,  1821,  when  it  is  7h.  12  m.  P.  M. 
mean  time  at  Greenwich,  what  is  the  apparent  time  at  Boston? 
Jins.  2  h.  31  m.  41  sec.  P.  M. 

4.  On  the  17th  of  September  1821,  when  it  is  10  h.  25  m. 
32  sec.  A.  M.  apparent  time  at  New  York,  what  is  the  mean 
time  at  Greenwich?  Ms.  3h.  16  m.  1  sec.  P.  M. 


262 


ASTRONOMY. 


PROBLEM  VII. 

To  find  the  Sun's  Longitude ,  Semidiameter,  and  Hourly  Mo¬ 
tion ,  and  the  apparent  Obliquity  of  the  Ecliptic ,  for  a  given  time , 
from  the  Tables . 

jpor  the  Longitude. 

When  the  given  time  is  not  for  the  meridian  of  Greenwich, 
reduce  it  to  that  meridian  by  prob.  V;  and  if  it  is  apparent  time 
reduce  it  to  mean  time,  by  the  last  problem. 

With  the  mean  time  at  Greenwich,  take  from  tables  IX,  X, 
XIII,  and  XIV,  the  quantities  corresponding  to  the  year,  month, 
day,  hour,  minute,  and  second,  and  find  their  sums.*  The  sum 
in  the  column  of  mean  longitudes  will  be  the  tabular  mean  longi¬ 
tude  of  the  sun;  the  sum  in  the  column  of  perigee,  will  be  the 
tabular  longitude  of  the  perigee;  and  the  sums  in  the  columns  I, 
II,  III,  and  N,  will  be  the  arguments  for  the  small  equations  of 
the  sun’s  longitude,  and  for  the  equation  of  the  equinoxes,  which 
forms  one  of  them. 

Subtract  the  longitude  of  the  perigee  from  the  sun’s  mean  lon¬ 
gitude,  borrowing  12  signs  when  necessary;  the  remainder  is  the 
sun’s  Mean  Anomaly.  With  the  mean  anomaly  take  the  equa¬ 
tion  of  the  Sun’s  centre  from  table  XV ;  and  with  the  arguments 
I,  II,  and  III,  take  the  corresponding  equations  from  table  XVi. 
The  equation  of  the  centre  and  the  three  other  equations,  added 
to  the  mean  longitude,  gives  the  sun’s  True  Longitude,  reckoned 
from  the  mean  equinox. 

With  the  argument  N,  take  the  equation  of  the  equinoxes,  or 
which  is  the  same  thing,  the  Nutation  in  Longitude,  from  table 
XVIII,  and  apply  it,  according  to  its  sign,  to  the  true  longitude 
already  found,  and  the  result  will  be  the  true  longitude,  from  the 
apparent  equinox, 

*  In  adding  quantities  that  are  expressed  in  signs,  degrees,  Stc.  reject  12 
or  24  signs,  when  the  sum  exceeds  either  of  these  quantities.  In  addin" 
any  arguments,  expressed  in  100,  or  1000,  Stc.  parts  of  the  circle,  when  th 
are  expressed  by  two  figures,  reject  the  hundreds  from  the  sum;  when 
three  figures,  the  thousands;  and  when  by  four  figures,  the  ten  thousand. 


ASTRONOMY. 


263 


For  the  Semidiameter  and  Hourly  Motion. 

With  the  sun’s  Mean  Anomaly,  take  the  Hourly  Motion  and 
Semidiameter,  from  tables  XI  and  XII. 

< 

For  the  Apparent  Obliquity  of  the  Ecliptic. 

To  the  Mean  Obliquity,  taken  from  table,  XVII,  apply,  ac¬ 
cording  to  its  sign,  the  Nutation  in  Obliquity,  taken  from  table 
XVIII,  with  the  argument  N,  and  the  result  will  be  the  Apparent 
Obliquity. 

Note.  In  the  Nautical  Almanac  the  Sun’s  Longitude  is  given 
for  each  day  in  the  year  at  apparent  noon;  and  the  Semidiameter 
and  Hourly  Motion  are  given  for  several  times  in  each  month.* 
Either  of  these  quantities  may  easily  be  found  for  any  interme¬ 
diate  times,  by  proportion.  The  Apparent  Obliquity  of  the 
Ecliptic  is  given  in  the  beginning  of  the  Almanac,  for  each  three 
months  in  the  year,  and  is  easily  estimated  for  any  intermediate 
time. 

Exam.  1.  Required  the  Sun’s  Longitude,  Hourly  Motion,  and 
Semidiameter,  and  the  Apparent  Obliquity  of  the  Ecliptic,  on 
the  18th  of  October,  1821,  at  3  h.  20  m.  18  sec.  P.  M.  mean 
time  at  Philadelphia. 

d.  h.  m.  sec. 

Mean  time  at  Philadelphia  Oct.  1821,  18  3  20  18 
Diff.  of  Long.  -  -  -  -  5  0  46 

Mean  time  at  Greenwich  -  18  8  21  4 


See  the  second  and  third  pages  of  table  LV. 


264 


ASTRONOMY. 


M.  Long. 

Long.  Perigee. 

I 

II 

III 

N 

1821 

9s  8°  48'  19" 

9*  7°  50'  43" 

920 

782 

260 

036 

Octob. 

8  29  4  54 

46 

250 

684 

468 

40 

18  d. 

16  45  22 

3 

574 

43 

29 

2 

8  h. 

19  43 

0 

11 

0 

0 

0 

21  m. 

52 

4  sec. 

0 

9  7  51  32 

755 

509 

757 

78 

6  24  59  10 

6  24  59  10 

i 

Eq.  Sun’s  cent. 

8  2  7 

9  17  7  38 

Mean  Anomaly. 

I 

4 

Sun’s  Hourly  Motion  2'  29" 

II 

10 

Sun’s  Semidiameter 

. 

16  5 

III 

6 

6  25  7  57 

M.  Obliq.  Ecliptic  1821,  23°  27 

''  46" 

Nutation. 

+  9 

Nutation 

- 

- 

+  8 

Sun’s  true  long. 

6  25  8  6 

Appar.  Obliquity 

-  23  27  54 

2.  Required  the  Sun’s  longitude,  hourly  motion,  and  semi¬ 
diameter,  and  the  obliquity  of  the  ecliptic,  on  the  19th  of 
August,  1821,  at  7  h.  4  m.  51  sec.  A.  M.  apparent  time  at  Phila¬ 
delphia.  Jins.  Sun’s  longitude  4s  26°  6'  43";  hourly  motion 
2'  25";  semidiameter  15'  51";  obliquity  of  the  ecliptic  23* 
27'  55". 

3.  Required  the  Sun’s  longitude,  hourly  motion,  and  semi- 
diameter,  and  the  obliquity  of  the  ecliptic,  on  the  21st  of  Feb¬ 
ruary,  1824,  at  9  h.  6  m.  17  sec.  P.  M.  mean  time  at  Philadel¬ 
phia.  Jins.  Sun’s  longitude  11s  2°  27'  46";  hourly  motion 
2'  31";  semidiameter  16'  12";  obliquity  of  the  ecliptic  23° 
27'  49". 


PROBLEM  VIII. 

The  Obliquity  of  the  Ecliptic  and  the  Sun's  longitude  being 
given ,  to  find  the  Right  Ascension  and  Declination. 

For  the  Right  Ascension. 

To  the  Cosine*  of  the  Obliquity,  add  the  Tangent  of  the  Lon¬ 
gitude,  and  reject  10  from  the  index;  the  resulting  logarithm  will 
be  the  the  Tangent  of  the  Right  Ascension  which  must  always 
be  taken  in  the  same  quadrant  as  the  longitude. 

*  By  the  terms  Sine,  Cosine,  &c.  are  here  meant  the  logarithmic  Sine,  Co¬ 
sine,  &c.  The  same  is  to  be  understood  when  the  terms  are  used  in  the 
rules  for  working  any  of  the  following  problems. 


ASTRONOMY. 


265 


For  the  Declination. 

To  the  Sine  of  the  Obliquity,  add  the  Sine  of  the  Longitude, 
and  reject  10  from  the  index;  the  resulting  logarithm  will  be  the 
Sine  of  the  Declination,  which  must  always  be  taken  out  less  than 
90°;  and  it  will  be  North  or  South,  according  as  the  sign  is  af¬ 
firmative  or  negative. 

Note.  The  Sun’s  right  ascension,  and  declination  are  given,  in 
the  Nautical  Almanac,  for  each  day  in  the  year.  See  table  LV. 

Exam.  1.  Given  the  obliquity  of  the  ecliptic  23°  27'  40",  and 
the  sun’s  longitude  125°  31'  25",  to  find  the  right  ascension  and 
declination. 


cos.  Obliquity 
tan.  Long. 

tan.  Right  Ascen. 

sin.  Obliquity 
sin.  Long. 

sin.  Decl. 


23°  27'  40" 

125  31  25  - 

127  53  30 

23°  27'  40" 
125  31  25  - 

18  54  23  N 


9.96253 
— 10.14635 

—  10.10888 

9.60002 
-  9.91055 

9.51057 


2.  The  obliquity  of  the  ecliptic  being  23°  27'  40",  what  is  the 
sun’s  right  ascension  and  declination,  when  his  longitude  is  35° 
19'  30"?  Ans.  Right  ascension  33“  1'  43",  and  declination  13° 
18'  32"  N. 


3.  Given  the  obliquity  of  the  ecliptic  23°  27'  50",  and  the 
sun’s  longitude  313°  36'  12";  what  is  the  right  ascension  and 
declination?  Ans.  Right  ascension  316°  4'  30",  and  declination 
16°  45'  29"  S. 


PROBLEM  IX. 

Given  the  Obliquity  of  the  Ecliptic  and  the  Sun's  Right  Ascen - 
5  ton,  to  find  the  Longitude  and  Declination . 

For  the  Longitude. 

To  the  arithmetical  complement  of  the  Cosine  of  the  Obliquity, 
35 


266 


ASTRONOMY. 


add  the  Tangent  of  the  Right  Ascension;  and  the  resulting  loga¬ 
rithm  will  be  the  Tangent  of  the  Longitude,  which  must  be  taken 
in  the  same  quadrant  as  the  Right  Ascension. 

For  the  Declination. 

To  the  Tangent  of  the  Obliquity,  add  the  Sine  of  the  Right 
Ascension,  and  reject  10,  from  the  index;  the  resulting  logarithm 
will  be  the  Tangent  of  the  Declination,  which  will  be  North  or 
South,  according  as  the  sign  is  affirmative  or  negative. 

Exam.  1 .  Given  the  Obliquity  of  the  ecliptic  23°  27'  50",  and 
the  sun’s  right  ascension  215°  12'  27";  what  is  the  longitude 
and  declination? 


cos  Obliquity 

23°  27'  50" 

Ar.  Co.  0.03748 

tan.  Right  Asc.  215°  12  27  - 

9.84857 

tan.  Long. 

217  34  5 

-  9.88605 

tan.  Obliquity 

23°  27'  50" 

9.63755 

sin.  Right  Asc.  215  12  27 

_  9.76083 

tan.  Declin. 

14  3  OS 

9.39838 

2.  When  the  obliquity  of  the  ecliptic  is  23°  27'  50",  and  the 
sun’s  right  ascension  53°  31'  20",  what  is  the  longitude  and  de¬ 
clination?  Ans.  Longitude  55°  51'  16",  and  declination  19° 
14'  24"  N. 

3.  Given  the  obliquity  of  the  ecliptic  23°  27'  40",  and  the 
sun’s  right  ascension  187°  15'  21";  required  the  longitude  and 
declination.  Ans.  Longitude  187°  54'  6",  and  declination  3° 
8'  15"  S. 

PROBLEM.  X 

The  Obliquity  of  the  Ecliptic  and  the  Sun's  Longitude  being 
given ,  to  find  the  angle  of  Position. 

To  the  Tangent  of  the  Obliquity,  add  the  Cosine  of  the  Lon- 


ASTRONOMY. 


S67 

gitude,  and  reject  10.  from  the  index;  the  resulting  logarithm  will 
be  the  Tangent  of  the  angle  of  Position,  which  must  always  be 
taken  less  than  90°. 

The  northern  part  of  the  circle  of  latitude  will  be  to  the  West 
or  East  of  the  Northern  part  of  the  circle  of  declination,  accor¬ 
ding  as  the  sign  of  the  tangent  of  the  Angle  of  Position  is  affirma¬ 
tive  or  negative. 

Exam.  1.  Given  the  obliquity  of  the  ecliptic  23°  27'  50°',  and 
the  sun’s  longitude  112°  19'  17",  to  find  the  angle  of  Position. 

tan.  Obliquity  23°  27'  50"  -  9.63755 

cos.  Long.  -  112  19  17  -  —  9.57956 

tan.  Angle  of  Posit.  9°  21'  41"  -  —  9.21711 

The  northern  part  of  the  circle  of  latitude  lies  to  the  east  of  the 
circle  of  declination. 

2.  Given  the  obliquity  of  the  ecliptic  23°  27'  50",  and  the 
sun’s  longitude  77°  47'  30";  what  is  the  angle  of  position?  Ans. 
5°  14'  40";  and  the  northern  part  of  the  circle  of  latitude  lies  to 
the  west  of  the  circle  of  declination. 

3.  When  the  obliquity  of  the  ecliptic  is  23°  27'  50",  and  the 
sun’s  longitude  225°  41'  12",  what  is  the  angle  of  position?  Ans. 
16°  15'  7";  and  the  northern  part  of  the  circle  of  latitude  lies  to 
the  east  of  the  circle  of  declination. 

PROBLEM  XI. 

To  find,  from  the  Tables ,  the  Moon's  Longitude ,  Latitude ,  Equa¬ 
torial  Parallax ,  Semidiameter ,  and  Hourly  Motions ,  in  Longitude 
and  Latitude,  for  a  given  time. 

When  the  given  time  is  not  for  the  meridian  of  Greenwich,  re¬ 
duce  it  to  that  meridian;  and  when  it  is  apparent  time,  reduce  it 
to  mean  time. 


268 


ASTRONOMY. 


With  the  mean  time  at  Greenwich,  take  out,  from  tables  XX, 
XXI,  XXII,  XXIII,  and  XXIV,  the  arguments,  numbered  1,  2, 
3,  &c.  to  20,  and  find  their  sums,  rejecting  the  ten  thousands,  in 
the  first  nine,  and  the  thousands,  in  the  others.  The  resulting 
quantities  will  be  the  arguments  for  the  first  twenty  equations  of 
Longitude. 

With  the  same  time,  and  from  the  same  tables,  take  out  the 
remaining  arguments  and  quantities,  entitled  Evection,  Anomaly, 
Variation,  Longitude,  Supplement  of  the  Node,  II,  V,  VI,  VII, 
VIII,  IX,  and  X;  and  add  the  quantities  in  the  column  for  the 
Supplement  of  the  Node. 

.  For  the  Longitude. 

With  the  first  twenty  arguments  of  longitude,  take,  from  tables 
XXV  to  XXX,  the  corresponding  equations,  and  place  their  sum 
in  the  column  of  Evection.  Then,  the  sum  of  the  quantities  in 
this  column  will  be  the  corrected  argument  of  Evection. 

With  the  corrected  argument  of  Evection,  take  the  Evection 
from  table  XXXI,  and  add  it  to  the  sum  of  the  preceding  equa¬ 
tions.  Place  the  resulting  sum,  in  the  column  of  Anomaly. 
Then,  the  sum  of  the  quantities  in  this  column  will  be  the  cor¬ 
rected  Anomaly. 

With  the  corrected  Anomaly,  take  the  Equation  of  the  Centre 
from  table  XXXII,  and  add  it  to  the  sum  of  all  the  preceding 
equations.  Place  the  resulting  sum,  in  the  column  of  Variation. 
Then,  the  sum  of  the  quantities  in  this  column  will  be  the  corrected 
argument  of  Variation. 

With  the  corrected  argument  of  Variation,  take  the  variation 
from  table  XXXIII,  and  add  it  to  the  sum  of  all  the  preceding 
equations;  the  result  will  be  the  sum  of  the  first  twenty  three 
equations  of  the  Longitude.  Place  this  sum  in  the  column  of 
Longitude.  Then,  the  sum  of  the  quantities  in  this  column  will 
be  the  Orbit  Longitude  of  the  Moon,  reckoned  from  the  mean 
equinox. 


ASTRONOMY. 


269 


Add  the  Orbit  Longitude,  to  the  Supplement  of  the  Node.  The 
result  will  be  the  argument  of  the  Reduction.  It  will  also  be  the 
1st  argument  of  Latitude. 

With  the  argument  of  Reduction,  take  the  Reduction  from  ta~ 
hie  XXXIV,  and  add  it  to  the  Orbit  Longitude.  Also,  with  the 
19th  argument,  which  is  the  same  as  argument  N,  for  the  Sun’s 
Longitude,  take  the  Nutation  in  Longitude,  from  table  XVIII, 
and  apply  it,  according  to  its  sign,  to  the  last  sum.  The  result 
will  be  the  Moon’s  true  Longitude  from  the  Apparent  equinox. 

For  the  Latitude . 

Place  the  sum  of  the  first  twenty  three  equations  of  Longitude, 
taken  to  the  nearest  minute,  in  the  column  of  Arg.  II.  Then  the 
sum  of  the  quantities  in  this  column  will  be  Arg.  II  of  Latitude, 
corrected.  The  Moon’s  true  Longitude  is  the  Illrd  argument  of 
Latitude.  The  20th  argument  of  Longitude  is  the  IVth  argu¬ 
ment  of  Latitude.  Convert  the  degrees  and  minutes,  in  the  sum 
of  the  first  twenty  three  equations  of  Longitude,  into  thousandth 
parts  of  the  circle,  by  taking  from  table  XXXVIII,  the  number 
corresponding  to  them.  Place  this  number  in  the  columns  V, 
VI,  VII,  VIII,  and  IX;  But  not  in  column  X.  Then  the  sums  of 
the  quantities  in  columns,  V,  VI,  VII,  VIII,  IX,  and  X,  rejecting 
the  thousands,  will  be  the  Vth,  Vlth,  Vllth,  Vlllth,  IXth,  and 
Xth  arguments  of  Latitude. 

With  the  sum  of  the  Supplement  of  the  Node,  and  the  Moon’s 
Orbit  Longitude,  which  is  Arg.  I  of  Latitude,  take  the  Moon’s 
distance  from  the  North  Pole  of  the  Ecliptic,  from  table  XXXV ; 
and  with  the  remaining  nine  arguments,  take  the  corresponding 
equations  from  tables  XXXVI,  XXXVII,  and  XXXIX.  The 
sum  of  these  ten  quantities  will  be  the  Moon’s  true  distance  from 
the  North  Pole  of  the  Ecliptic.  The  difference  between  this 
distance  and  90°,  will  be  the  Moon’s  true  latitude;  whiclrwillbe 
North  or  South,  according  as  the  distance  is  less  or  greater 
than  90°. 


§70 


ASTRONOMY. 


For  the  Equatorial  Parallax. 

With  the  corrected  arguments,  Evection,  Anomaly  and  Va¬ 
riation,  take  the  corresponding  quantities  from  tables  XL,  XLI, 
and  XLII.  Their  sum  will  be  the  Equatorial  Parallax. 

For  the  Semidiameter . 

With  the  Equatorial  Parallax,  take  the  Moon’s  Semidiameter 
from  table  XLIV. 

For  the  Hourly  Motion  in  Longitude. 

With  the  arguments,  2,  3,  4,  and  5,  of  Longitude,  rejecting 
the  two  right  hand  figures  in  each,  take  the  corresponding  equa¬ 
tions  fromtable  XLVI.  Also  with  the  correct  argument  of  Evec¬ 
tion,  take  the  equation  from  table  XLVII. 

With  the  Sum  of  the  preceding  equations  at  top,  and  the  cor¬ 
rect  anomaly  at  the  side,  take  the  equation  from  table  XLVIII. 
Also  with  the  correct  anomaly  take  the  equation  from  table 
XLIX. 

With  the  Sum  of  all  the  preceding  equations  at  the  top,  and 
the  correct  argument  of  Variation,  at  the  side,  take  the  equation 
from  table  L.  With  the  correct  argument  of  Variation,  take  the 
equation  from  table  LI.  And  with  the  argument  of  Reduction, 
take  the  equation  from  table  LII.  These  three  equations  added  to 
the  sum  of  all  the  preceding  ones,  will  give  the  Moon’s  Hourly 
Motion  in  Longitude. 

For  the  Hourly  Motion  in  Latitude. 

With  the  1st  and  2d  arguments  of  Latitude,  take  the  corres¬ 
ponding  quantities  from  tables  LIII  and  LIV,  and  find  their  sum, 
attending  to  the  signs.  Then  32'  56"  :  the  moon’s  true  hourly 
motion  in  Longitude  :  :  this  sum  :  the  moon’s  true  hourly  motion 
in  Latitude.  When  the  sign  is  affirmative  the  moon  is  tending 
North;  and  when  it  is  negative,  she  is  tending  South. 


ASTRONOMY. 


274 

Exam.  1.  Required  the  moon’s  longitude,  latitude,  equatorial 
parallax,  semidiameter,  and  hourly  motions  in  longitude  and  lati¬ 
tude,  on  the  6th  of  August  1821,  at  8h.  46  m.  27  sec.  A.  M. 
mean  time  at  Philadelphia. 

d.  h.  m.  sec. 

Mean  time  at  Philadelphia,  August,  5  20  46  27 
DifT.  of  Long.  -  5  0  46 


Mean  time  at  Greenwich,  August,  6  1  47  13 


272 


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273 


30 


An.  4  13  50  Moon’s  Eq.  Par.  54  31 

- Moon’s  Semidiameter  14'  51".  As  32'  56" :  3/  3"  : :  —  O'  49"  :  — 0'  45"  =  Moon’s  hourly  motion 

Sum  6  1  31  in  latitude,  tending  South. 

Yar.  30  31 


ASTRONOMY. 


27* 

2.  Required  the  Moon’s  longitude,  latitude,  equatorial  parallax, 
semidiameter,  and  hourly  motions  in  longitude  and  latitude,  on 
the  27th  of  April,  1821,  at  9  h.  43  m.  30  sec.  P.  M.  mean  time 
at  Baltimore.  Jins.  Long.  11s  13°  32'  13";  lat.  6'  57"  N,  equat. 
par.  60'  0";  semidiam.  16'  21";  hor.  mot.  in  long.  36'  11";  and 
hor.  mot.  in  lat.  3'  14",  tending  north. 

3.  What  will  be  the  Moon’s  longitude,  latitude,  equatorial  pa¬ 
rallax,  semidiameter,  and  hourly  motions  in  longitude  and  latitude, 
on  the  19th  of  August,  1822,  at  5h.  56  m.  14  sec.  P.  M.  mean 
time  at  Philadelphia?  Jins.  Long.  6s  3°  7'  12";  lat.  3°  54  35"  S  : 
equat.  par.  56'  19";  semidiam.  15'  21";  hor.  mot.  in  long.  32' 
7";  and  hor.  mot.  in  lat.  2'  1",  tending  south. 

PROBLEM  XII. 

To  find  the  Moon’s  Longitude ,  Latitude ,  Hourly  Motions ,  Equa¬ 
torial  Parallax ,  and  Semidiameter ,  for  a  given  Time ,  from  the 
Nautical  Almanac. 

Reduce  the  given  time  to  Apparent  time  at  Greenwich.  Then, 
For  the  longitude. 

Take  from  the  Nautical  Almanac,  the  two  longitudes,  for  the 
noon  and  midnight,  or  midnight  and  noon,  next  preceding  the 
time  at  Greenwich,  and  also  the  two  immediately  following  these, 
and  set  them  in  succession,  one  under  another.  Then,  having 
regard  to  the  signs,  subtract  each  longitude,  from  the  next  follow¬ 
ing  one,  and  the  three  remainders  will  be  the  first  differences. 
Call  the  middle  one  A.  Subtract  each  first  difference  from  the 
following,  for  the  second'  differences.  Take  the  half  sum  of  the 
second  differences  and  call  it  B. 

Call  the  excess  of  the  given  time  at  Greenwich,  above  the 
time  of  the  second  longitude,  T.  Then  12  h  :  T  :  :  A  :  fourth 
ter)n ,  which  must  have  the  same  sign  as  A. 

With  the  time  T  at  the  side,  take  from  table  LVI,  the  quantities 
corresponding  to  the  minutes,  tens  of  seconds,  and  seconds  of  B, 
at  the  top,  the  sum  of  these,  with  a  contrary  sign  to  that  of  B, 
will  be  the  correction  of  second  differences. 


ASTRONOMY. 


275 

The  sum  of  the  second  longitude,  the  fourth  term,  and  the  cor¬ 
rection  of  second  differences,  having  regard  to  the  signs,  will  be 
the  required  longitude. 

For  the  Hourly  Motion  in  Longitude. 

To  the  logistical  logarithm  of  of  T,  add  the  logistical  loga¬ 
rithm  of  B,  and  find  the  quantity  corresponding  to  the  sum. 
Call  this  quantity  C,  and  prefix  to  it  the  same  sign  as  that  of  B. 

Or  C  may  be  found  without  logarithms;  thus,  12  h  :  T  :  : 
B  :  C. 

Divide  the  sum  of  A,  \  B  with  its  sign  changed,  and  C,  by  12, 
and  the  quotient  will  be  the  required  hourly  motion  in  longitude. 

For  the  Latitude. 

Prefix  to  north  latitudes  the  affirmative  sign,  but  to  south  lati¬ 
tudes  the  negative  sign,  and  then  proceed  in  the  same  manner  as 
for  the  longitude.  The  resulting  latitude  will  be  north  or  south) 
according  as  its  sign  is  affirmative  or  negative. 

Note.  The  Moon’s  Declination  may  be  found  in  the  same 
manner. 

For  the  Hourly  Motion  in  Latitude. 

With  T,  and  the  values  that  A  and  B  have,  in  finding  the  lati¬ 
tude,  find  the  hourly  motion  in  latitude,  in  the  same  manner  as 
directed  for  finding  the  hourly  motion  in  longitude.  When  the 
resulting  hourly  motion  in  latitude  is  affirmative ,  the  moon  is 
tending  north ;  and  when  it  is  negative,  she  is  tending  south. 

For  the  Semidiameter  and  Equatorial  Parallax. 

The  Moon’s  semidiameter  and  equatorial,  horizontal  parallax 
may  be  taken  from  the  Nautical  Almanac  with  sufficient  accuracy 
by  simply  proportioning  for  the  odd  time  between  noon  and  mid¬ 
night  or  midnight  and  noon. 

Exam.  1.  Required  the  Moon’s  longitude,  latitude,  equatorial 
parallax,  semidiameter,  and  hourly  motions  in  longitude  and  lati- 


ASTRONOMY 


276 


tude,  from  the  Nautical  Almanac,  on  the  6th  of  August,  1821,  at 
8h.  40  m.  54  sec.  apparent  time  at  Philadelphia. 

d.  h.  m.  sec. 

Appar.  time  at  Philadelphia,  August  5  20  40  54 
Diff.  of  Long.  -  5  0  46 


Appar.  time  at  Greenwich,  August  6  1  41  40 
For  the  Longitude  and  Hourly  Motion  in  Longitude . 

Mean  of 


Longitudes 

1st  Diff. 

2d  Diff. 

2d  Diff. 

5th  midn. 
6th  noon 
6th  midn. 
7th  noon 

7s  12°  8' 55" 
7  18  7  55 

7  24  9  18 

8  0  13  38 

5°  59'  0" 
A.  6  1  23 

6  4  20 

2' 23" 
2  57 

B.  +  2' 40" 

T 

h.  b.  m. 
12:1  41 

A 

sec. 

40  6°  1'  23"  :  51'  l."7,  fourth  term. 

Second  Longitude  - 

Fourth  term  - 

Cor.  2d  diff.  from  tab.  LVI 

7s  18° 

7'  55" 

51  1.7 

—  9.7 

Moon’s  true  Longitude 

7  18 

58  47 

1  nn 

12  A 

B  - 

-  -  8  m.  28  sec. 

+  2  40 

L.  L. 

L.  L. 

8504 

13522 

C 

-  -  +0 

23 

22026 

A 

6°  r 

23" 

i  B,  sign  changed,  -  —  1  20 

C  -  -  -  +  0  23 


12)6  0  26 


Hor.  mot.  in  long. 


30'  2". 2 


ASTRONOMY. 


277 


For  the  Latitude  axd  Hourly  Motion  in  Latitude. 


Mean  of 


Latitudes 

1st  Diff. 

2d  Diff. 

2d  Diff. 

5th  midn. 
6th  noon 
6th  midn. 
7  th  noon 

—  4”  50'  53" 

—  5  1  55 

—  5  9  43 

—  5  14  18 

—  11'  2" 
A.  —  7  48 
—  4  35 

+  3'  14" 
+  3  13 

B.  +  3'  13".5 

T  A 

h.  h.  m.  sec. 

12  : 1  41  40  :  :  —  7'  48"  :  —  1'  6".l,  fourth  term. 


Second  Latitude  -  -  — 5°  1'  55" 

Fourth  term  -  -  -  —  1  6.1 

Cor.  2d  diff.  -  -  -  —  11.7 


Moon’s  true  Latitude 

T  8  m.  28  sec. 
B  +3  13". 5 


5  3  13  S. 

L.  L.  8504 

L.  L.  12696 


C  +  0  27  21200 

A  -  -  -  -  —7'  48” 

§  B,  sign  changed  -  —  1  37 

C  +0  27 

12)— 8  58 

Hor.  mot.  in  lat.  -  — 0'  44".8,  tending  south. 

Moon’s  semidiam.  from  N.  Aim.  14'  53" 
do.  eq.  parallax  -  -  -  54  32. 


Note.  The  quantities  found  in  this  example,  from  the  Nautical 
Almanac,  are  for  the  same  time  as  those  found  in  example  1st  of 
the  last  problem,  from  the  tables  in  this  work.  It  may  be  seen 
that  there  is  not  much  difference  in  them. 

2.  Required  the  Moon’s  longitude,  latitude,  equatorial  parallax, 
semidiameter,  and  hourly  motions  in  longitude  and  latitude,  on  the 
21st  of  August  1821,  at  16  h.  20  m.  33  sec.  apparent  time  at 


S78 


ASTRONOMY. 


Greenwich.  Jins.  Long.  2s  23°  7'  43";  lat.  5°  O'  36"  N;  equat. 
par.  57'  57";  semidiam.  15' 49";  hor.  mot.  in  long.  34'  3".8; 
and  hor.  mot.  in  lat.  O'  59".4,  tending  south. 

3.  What  were  the  Moon’s  longitude,  latitude,  equatorial  pa¬ 
rallax,  semidiameter,  and  hourly  motions  in  longitude  and  lati¬ 
tude,  on  the  14th  of  August  1821,  at  2  h.  8  m.  2  sec.  P.  M.  ap¬ 
parent  time  at  Philadelphia?  Jins.  Long.  11s  7°  43'  8";  lat.  0° 
17'  6"N;  equat.  par.  59'  47";  semidiam.  16  19";  hor.  mot.  in 
long.  36'  4".2;  and  hor.  mot.  in  lat.  3'  19". 3,  tending  north. 

PROBLEM  XIII. 

The  Moon's  Equatorial  Parallax,  and  the  Latitude  of  a  Place 
being  given ,  to  find  the  Reduced  Parallax  and  Latitude . 

With  the  Latitude  of  the  place,  take  the  Reductions  from 
table  XLIII,  and  subtract  them  from  the  Parallax  and  Latitude. 

Exam.  1.  Given  the  equatorial  parallax  54'  31",  and  the 
latitude  of  Philadelphia  39c  27'  N,  to  find  the  reduced  parallax 
and  latitude. 


Equatorial  parallax 

54'  31" 

Reduction  - 

5 

Reduced  Parallax 

54  26 

Latitude  of  Philadelphia 

29°  57'  N. 

Reduction  - 

11 

Reduced  Lat.  of  Philadelphia 

39  46  N. 

2.  Given  the  equatorial  parallax  60°  0",  and  the  latitude  of 
Boston  42°  23'  N.  to  find  the  reduced  parallax  and  latitude.  Jins. 
Reduced  par.  59'  55",  and  reduced  lat.  42°  12'  N. 

3.  Given  the  equatorial  parallax  57'  21",  and  the  latitude  of 
Charleston  32°  50'  N.  to  find  the  reduced  parallax  and  latitude. 
Jins .  Reduced  par.  57'  18",  and  reduced  lat.  32°  40' N. 


ASTRONOMY. 


279 


PROBLEM  XIV. 


To  find  the  Mean  Right  Ascension  and  Declination ,  or  Lon¬ 
gitude  and  Latitude  of  a  Star  for  a  given  Time ,  from  the  tables. 


Take  the  difference  between  the  time  for  which  the  table  is 
constructed  and  the  given  time,  and  multiply  the  annual  variation, 
by  the  number  of  years  in  this  difference;  the  product  will  be  the 
number  of  years  in  this  difference;  the  product  will  be  the  va¬ 
riation  for  the  years.  Reduce  the  odd  time  to  days.*  Then, 
365  days  :  number  of  days  :  :  annual  variation  :  proportional  part. 
This  proportional  part,  added  to  the  variation  for  the  years,  will 
be  the  whole  variation,  which  applied  to  the  quantity  given  in 
the  table,  with  its  proper  sign,  when  the  given  time  is  after  the 
time  for  which  the  table  is  constructed,  but  with  a  contrary  sign 
when  it  is  before,  will  give  the  required  quantity. 

Exam.  1.  Required  the  mean  right  ascension  and  declination 
of  Regulus ,  on  the  15th  of  June  1821. 

Mean  right  ascen.  begin,  of  1820,  table  III,  149°  41'  39" 

Var.  for  1  yr.  166  d.  -  -  -  -  -j-  1  10 

Mean  right  ascen.  required  -  -  149  42  49 


Mean  declin.  begin,  of  1820, 
Var.  for  1  yr.  166  d. 

Mean  declin.  required 


12°  50'  36"  N 
—  25 

52  10  11 N. 


2.  Required  the  mean  longitude  and  latitude  of  Regulus  on  the 
15th  of  June,  1821. 


Mean  longitude,  begin,  of  1810,  table  VIII,  4s  27°  11'  18" 
Var.  for  11  y.  166  d.  ...  +  9  33 

Mean  long,  required  -  -  -  -  4  27  20"  51 


*  This  when  the  given  time  is  after  the  time  for  which  the  table  is  con¬ 
structed,  may  be  done  very  simply  by  taking  from  table  Yl,  the  number  of 
days  corresponding  to  the  month,  and  adding  to  it  the  odd  days* 


280 


ASTRONOMY. 


Mean  latitude,  begin,  of  1810  0°  27'  36"  N 

Var.  for  11  y.  1G6  d.  -  4.  2 

Mean  latitude,  required  -  ~  -  0  27  38  N. 

3.  Required  the  mean  right  ascension  and  declination  of  £ 
Tauri ,  on  the  6th  of  November  1822.  Ans .  Mean  right  ascen. 
78°  46'  29'"  and  mean  declin.  28°  26'  53"  N. 

4.  What  will  be  the  mean  longitude  and  latitude  of  /3  Tauri , 
on  the  6th  of  November  1822?  Ans .  Mean  long.  2s  20°  5  59", 
and  mean  Iat.  5°  22'  27"  N. 

PROBLEM  XV. 

To  find  the  Aberration  of  a  Star ,  in  right  Ascension  and  Decli¬ 
nation,  for  a  given  Day . 

Find  the  mean  right  ascension  and  declination  of  the  star,  for 
the  given  time  by  the  last  problem.  Also  find  the  sun’s  true  lon¬ 
gitude  for  noon  of  the  given  day  by  prob.  VII,  or  take  it  from 
the  Nautical  Almanac. 

Designate  the  sun’s  longitude  by  0,  the  right  ascension  of  the 
star  by  A,  and  the  declination  by  D. 

With  the  argument  0,  take  the  quantity  x  from  table  LX;  add 
it  to  ©j  attending  to  the  sign  and  from  the  sum,  subtract  A. 
Then, 

For  the  Aberration  in  Right  Ascension. 

With  argument  0,  take  from  table  LX,  the  log.  a ,  with  its 
proper  sign,  and  to  it.  add  the  Cosine  of  (0  4-  x  —  A),  and  the 
arithmetical  complement  of  the  Cosine  of  D,  rejecting  the  tenp  in 
the  index  of  the  sum.  The  natural  number,  corresponding  to  the 
resulting  logarithm  will  be  the  aberration  in  right  ascension,  to  be 
applied  to  the  mean  right  ascension. 

For  the  Aberration  in  Declination. 

Add  together  the  log.  a ,  the  Sine  of  (0  +  x  —  A),  and  the  Sine 
of  D,  and  reject  the  tens  in  the  index  of  the  sum.  Take  the  natural 


ASTRONOMY. 


281 

number,  corresponding  to  the  sum,  and  call  it  m.  With  the  ar¬ 
guments  ©  4-  D  and  e —  D,  or  when  the  declination  is  south, 
with  these  arguments,  each  increased  by  VI  signs,  take  the  cor¬ 
responding  quantities  from  table  LXI.  The  sum  of  these  quan¬ 
tities,  and  m,  giving  attention  to  the  signs,  will  be  the  aberration 
in  declination,  to  be  applied  to  ihe  mean  declination. 

Exam.  1.  What  are  the  aberrations  in  right  ascension  and  de¬ 
clination,  of  Regains,  on  the  15th  of  June,  1821,  the  sun’s  lon¬ 
gitude  on  that  day,  being  2s  23°  58'? 

A  =  149°  42'  49"  =  mean  right  ascen.  of  Regulus. 

D  =  12  50  1 1  N  =  mean  declin.  do, 

©  =  2s  23°  58'  =  sun’s  longitude. 


© 

- 

2s 

23° 

58' 

Xy  from  table  LX, 

4- 

0 

30 

©  +  x 

- 

-  2 

24 

28 

A 

- 

4 

29 

43 

©  4-  x — A 

- 

9 

24 

45  = 

294°  45' 

log.  a  from  tab.  LX, 

_ 

. 

—  1.3061 

cos.  (e|i- 

-A) 

294c 

‘  45' 

9.6.219 

cos  D 

12 

50  Ar.  Co.  0.0110 

Aber.  in  right 

ascen.  - 

-8". 

69 

- 

- 0.9390 

log.  a  - 

. 

. 

. 

- 

—  1.3061 

sin  (©  4-  x  — 

A)  - 

294 

:°  45' 

-  —9.9581 

sin  D 

12 

50 

N.  - 

9.3466 

m 

- 

+ 

4".08 

0.6106 

Arg.  (©  -f  D) 

=  3s  6° 

48', 

gives 

- 

4-  0".4S 

Arg.  v® —  D) 

=  2  11 

8, 

gives 

- 

-  —  1.30 

m  - 

- 

- 

- 

- 

4-  4.08 

Aber.  in  declination 

. 

_ 

-  4-  3.26 

2.  Required  the  aberrations  in  right  ascension  and  declination, 
of  Antares ,  on  the  11  th  of  March,  1821,  the  sun’s  longitude  being 

37 


\ 


383 


ASTRONOMY. 


11s  20°  38'.  Jins.  Aber.  in  right  ascen.  -f-  5". 43,  in  declin, 
—  0".74. 

3.  On  the  6th  of  November  1822,  the  sun’s  longitude  will  be 
7s  13°  32';  what  will  be  the  aberrations  in  right  ascension  and  de¬ 
clination  of  /3  Tamil  Jins .  Aber.  in  right  ascension  -f  18". 54, 
and  in  declination  4-  0".15. 

PROBLEM  XVI. 

To  find  the  Mutations  of  a  Star  in  Right  Ascension  and  Decli¬ 
nation,  for  a  given  Time. 

Find  the  Supplement  of  the  Mooh’s  Node,  from  tables  XX, 
XXI  and  XXII,  and  subtract  it  from  12s  0°  7';  the  remainder 
will  be  the  Mean  Longitude  of  the  Moon’s  Ascending  Node. 

Designate  the  right  ascension  of  the  body  by  A,  the  declination 
by  D,  and  the  mean  longitude  of  the  moon’s  node  by  N. 

With  the  argument  N,  take  the  quantity  B,  from  table  LXII; 
add  it  to  N,  attending  to  the  sign,  and  from  the  sum,  subtract  A. 

For  the  Mutation  in  Right  Ascension. 

With  the  argument  N,  take  from  table  LXII,  the  log.  5,  with 
its  propersign,  and  to  it,  add  the  Cosine  of  (N  4-  B  —  A),  and  the 
Tangent  of  D,  marking  it  negative,  when  the  declination  is  south, 
and  reject  the  tens  in  the  index  of  the  sum.  Apply  the  natural 
number  corresponding  to  the  sum,  to  a  quantity,  taken  fromtable 
LXIII,  with  the  argument  N,  and  the  result  will  be  the  nutation 

in  right  ascension,  to  be  applied  to  the  mean  right  ascension. 

\ 

For  the  Mutation  in  Declination. 

To  the  log.  b ,  add  the  Sine  of  (N  4-  B  —  A),  or  when  the  de¬ 
clination  is  south,  the  Sine  of  (N  4.  B  —  A  -f  VIs)  rejecting  the 
tens  in  the  index,  and  the  natural  number  corresponding  to  the 
sum,  will  be  the  nutation  in  declination,  to  be  applied  to  the 
mean  declination. 

Exam.  1.  Required  the  nutations  in  right  ascension  and  de¬ 
clination,  of  Rigel,  on  the  19th  of  July  1825, 


ASTRONOMY. 


£88 


By  prob.  XIV,  A  =  76°  33'  8"  and  D  =  8°  24'  36"  S. 

Supp.  of  Node 


1825 

3s 

0° 

25' 

July  -  -  - 

- 

9 

35 

19  d. 

- 

57 

3 

10 

57 

12 

0 

7 

N 

8 

19 

10 

B,  from  tab.  LXII. 

— 

3 

35 

N  +  B  - 

8 

15 

35 

A 

-  2 

16 

33 

N  +  B  — A  - 

5 

29 

2  =  179°  2 

log.  6,  from  tab.  LXII, 

m 

—  0.8623 

cos.  (N  +  B  —  A) 

179°  2' 

—  9.9999 

tan.  D 

8  25 

- 

—  9.1702 

nat.  numb. 

—  1".0S 

—  0.0324 

From  tab.  LXIII. 

+  16.25 

Nut.  in  right  ascen. 

+  15,17 

log.  b. 

_ 

—  0.8623 

sin.  (N  +  B  —  A  +  VIs)  359' 

3  2' 

—  8.2271 

Nut  in  declin. 

+  d".i2 

+  9.0894 

2.  Required  the  nutations,  in  right  ascension  and  declination, 
of  Antares  on  the  11th  of  March,  1821,  Jbis.  Nut.  in  right 
ascen.  +  3". 70,  and  in  declin,  +  9".23. 

3.  What  will  be  the  nutations,  in  right  ascension  and  declina¬ 
tion  of  a  Tauri ,  on  the  6th  of  November,  1822.  Jins,  Nut,  in 
right  ascen.  +  14".61,  and  in  declin.  +  7".3. 


284 


ASTRONOMY. 


PROBLEM  XVII. 

To  find  the  Aberrations  of  a  Star  in  Longitude  and  Latitude, 
for  a  given  Time. 

Designate  the  sun’s  longitude  on  the  given  day  by  L.  the  mean 
longitude  of  the  star  by  L',  and  the  mean  latitude  by  a.  Then, 

For  the  Aberration  in  Longitude. 

Add  together  the  constant  logarithm,  —  1.30649,  the  Cosine 
of  (L  — -L'),  and  the  arithmetical  complement  of  the  Cosine  of  a, 
rejecting  the  tens  in  the  index  of  the  sum.  The  natural  number, 
corresponding  to  this  sum,  will  be  the  aberration  in  Longitude, 
to  be  applied  to  the  mean  longitude. 

For  the  Aberration  in  Latitude. 

Add  together  the  constant  logarithm,  —  1.30649,  the  Sine  of 
(L  —  L'),  and  the  Sine  of  a,  rejecting  the  tens  in  the  index,  and 
the  natural  number  corresponding  to  the  sum,  will  be  the  aberra¬ 
tion  in  latitude,  to  be  applied  to  the  mean  latitude. 

Exam.  1.  Required  the  aberrations  in  longitude  and  latitude, 
of  Sirius ,  on  the  20th  of  July,  1821,  the  sun’s  longitude  being 
3s  27  21'. 

Byprob.  XIV,  L'  =  3*  11°  37'  51"  and  a  =  39°  22'  31"  S. 


cos.  (L  —  L')  15°  43' 

—  1.30649 
9.98345 

cos.  a  39  23  Ar.  Co. 

0.11187 

Aber,  in  long.  —  25".22 

—  1.40181 

sin  (L  —  L')  15°  43'  - 

—  1.30649 
9.43278 

sin  a  39  23  - 

-  9.80244 

Aber.  in  lat.  —  3". 48 

-  —  0.54171 

2.  On  the  25th  of  April,  1822,  the  sun’s  longitude  will  be 
Is  4°  41';  what  will  be  the  aberrations  in  longitude  and  latitude 


ASTRONOMY.  28 5 

of  Regulus,  at  that  time?  Ans .  Aber.  in  long.  +  7". 81,  and  in 
lat.  +  O'M  5. 

3.  Required  the  aberrations  in  longitude  and  latitude  of  Vir- 
ginis ,  on  the  10th  of  August,  1821.  Jins.  Aber.  in  long. — • 
16". 14,  and  in  lat.  q-  0".15. 

PROBLEM  XVIII. 

To  find  the  Nutation  of  a  Body  in  Longitude f 

Find  the  mean  longitude  of  the  moon’s  ascending  node,  as  in 
prob.  XVI,  and  to  its  Sine,  add  the  constant  logarithm  — 
1.25396,  rejecting  the  tens  in  the  index.  The  natural  number 
corresponding  to  the  sum,  will  be  the  nutation  in  longitude,  to  be 
applied  to  the  mean  longitude. 

Exam.  1.  Required  the  nutation  in  longitude  of  Sirius,  on  the 
20th  of  July,  1821. 

The  mean  longitude  of  the  moon’s  ascending  node  at  the  given 
time  is  11s  6°  29'. 

—  1.26396 

sin.  long,  of  node  336°  29'  —  9.60099 
Nut.  in  long,  q-  7". 16  0.85495 

2.  Required  the  nutation  in  longitude  of  p  Virginis ,  on  the  10th 
of  August,  1821.  Jins .  q.  7 ".48. 

3.  What  will  be  the  nutation  in  longitude  of  Regulus ,  on  the 
25th  of  April,  1822?  Jins,  q-  11". 12. 

PROBLEM  XIX. 

The  Obliquity  of  the  Ecliptic  and  the  Right  Ascension  and  De¬ 
clination  of  a  Body  being  given ,  to  find  the  longitude  and  La¬ 
titude. 

4 

Designate  the  obliquity  of  the  ecliptic  by  E.  To  the  Tangent 
of  the  declination  marked  negative  when  the  declination  is  south, 
add  the  arithmetical  complement  of  the  Sine  of  the  right  ascen- 


286 


ASTRONOMY. 


sion;  the  result  will  be  the  Tangent  of  an  arc,  which,  call  B. 
The  arc  B  must  be  taken  acccording  to  the  sign,  but  always  less 
than  180°. 

4 

For  the  Longitude. 

Add  together  the  Cosine  of  the  difference  between  B  and  E, 
the  Tangent  of  the  right  ascension,  and  the  arithmetical  comple¬ 
ment  of  the  Cosine  of  B,  rejecting  10  from  the  index;  the  result 
will  be  the  tangent  of  the  longitude,  which  must  be  taken  ac¬ 
cording  to  the  sign,  observing  also  that  the  longitude  and  right  as¬ 
cension  are  always,  either  both  between  90°  and  270°,  or  reckon¬ 
ing  in  the  order  of  the  signs,  both  between  270°  and  90°. 

For  the  Latitude . 

To  the  Tangent  of  the  difference  between  B  and  E,  which 
must  be  marked  negative,  not  only  when  the  difference  is  greater 
than  90°,  but  also  when  E  is  greater  than  B,  add  the  Sine  of  the 
longitude,  rejecting  10  from  the  index;  the  result  will  be  the 
Tangent  of  the  latitude,  which  must  always  be  taken  less  than 
90°,  and  will  be  north  or  south ,  according  as  the  sign  is  affirma¬ 
tive  or  negative. 

Note.  When  the  mean  obliquity  of  the  ecliptic  and  the  mean 
right  ascension  and  declination  are  used,  the  results  will  be  the 
mean  longitude  and  latitude.  But  when  the  apparent  obliquity 
of  the  ecliptic,  found  by  prob.  VII,  and  the  apparent  right  ascen¬ 
sion  and  declination,  found  by  applying  to  the  mean  right  ascen¬ 
sion  and  declination,  the  aberrations  and  nutations,  obtained  by 
problems  XV  and  XVI,  are  given,  the  results  will  be  the  apparent 
longitude  and  latitude. 

Exam.  1.  On  the  10th  of  April,  1821,  the  mean  right  ascen¬ 
sion  of  Arcturus  was  211°  52'  37",  the  mean  declination  20°  7' 
4"  N,  and  the  mean  obliquity  of  the  ecliptic  23°  27'  46".  What 
were  its  longitude  and  latitude. 


ASTRONOMY. 


287 


tan.  Declin.  - 

20°  7'  4"  N  - 

9.56384 

sin.  Right  Asc. 

211  52  37  Ar.  Co. 

—  0.27729 

tan.  B 

-  145  15  13  - 

—  9.84113 

E 

23  27  46 

cos.  (B~E) 

121  47  27 

—  9.72166 

tan.  Right  Asc. 

211  52  37  - 

9.79371 

cos.  B 

-  145  15  13  Ar.  Co. 

—  0.08529 

tan.  Long. 

201  44  16 

9.60066 

tan.  (B  a?  E) 

141°  47' 27"  - 

-10.20774 

sin.  Long. 

201  44  16 

-  9.56863 

tan.  Lat. 

30  51  37 N 

9.77637 

2.  Given  the  obliquity  of  the  ecliptic  23°  27'  47",  the  right 
ascension  of  Rigel ,  76°  28' 21",  and  the  declination  8°  25'  2"S, 
on  the  1st  of  January,  1820,  to  find  the  longitude  and  latitude. 
Ans.  Long.  74°  18'  51",  and  lat.  31°  8'  45"  S. 

3.  On  the  first  of  January,  1821,  the  right  ascension  of  Pro- 
cyon  was  112°  28'  49",  the  declination  5°  40'  35"  N,  and  the 
obliquity  of  the  ecliptic  23°  27'  46".  What  were  its  longitude 
and  latitude?  Ms.  113°  18'  55",  and  15°  59'  0"S> 

PROBLEM  XX. 

The  Obliquity  of  the  Ecliptic ,  and  the  Longitude  and  Latitude 
of  a  Body  being  given ,  to  find  the  Right  Ascension  and  Declination . 

Designate  the  obliquity  of  the  ecliptic  by  E.  To  the  Tangent 
of  the  latitude,  marked  negative  when  the  latitude  is  south,  add 
the  arithmetical  complement  of  the  Sine  of  the  longitude;  the  re¬ 
sult  will  be  the  tangent  of  an  arc,  which  call  B.  The  arc  B  must 
be  taken,  according  to  the  sign,  but  always  less  than  180°. 

For  the  Right  Ascension. 

Add  together  the  Cosine  of  the  sum  of  B  and  E,  the  Tangent  of 
the  longitude,  and  the  arithmetical  complement  of  the  Cosine  of  B, 
rejecting  10  from  the  index;  the  result  will  be  the  Tangent  of  the 


2SH 


ASTRONOMY. 


right  ascension,  which  must  be  taken  according  to  the  sign, 
observing  also  that  the  right  ascension  and  longitude  are  always, 
either  both  between  90°  and  270°,  or  reckoning  in  the  order  of  the 
signs,  both  between  270°  and  90°. 

For  the  Declination. 

To  the  Tangent  of  the  sum  of  B  and  E,  add  the  Sine  of  the 
right  ascension,  rejecting  10  from  the  index;  the  result  will  be 
the  Tangent  of  the  declination,  which  must  always  be  taken  less 
than  90°,  and  will  be  north  or  south ,  according  as  the  sign  is  af¬ 
firmative  or  negative. 

Note.  The  quantities  found  will  be  mean  or  apparent,  ac- 
Wording  as  the  given  ones  are  mean  or  apparent. 

Exam.  1.  Given  the  obliquity  of  the  ecliptic  23°  27'  46",  the 
longitude  of  Arcturus  20V  44'  16",  and  the  latitude  30°  5T  37" 
N,  to  find  the  right  ascension  and  declination. 


tan.  Lat. 
sin.  Long.  - 

30°  51'  37"  N 

201  44  16  Ar.  Co. 

9.77637 
—  0.43137 

tan.  B  - 
E  - 

121  47  28 

23  27  46 

- 10.20774 

cos.  (B  +  E) 
tan.  Long, 
cos.  B 

145  15  14  - 

201  44  16 

-  121  47  28  Ar.  Co. 

—  9.91471 
9.60066 

—  0.27834 

tan.  Right  Asc. 

211  52  36 

9.79371 

tan.  (B  -f-  E) 
sin.  Right  Asc. 

145°  15'  14"  - 
211  52  36 

—  9.84113 

—  9.72271 

tan.  Declin. 

1 

£ 

o 

9.56384 

2.  Given  the  obliquity  of  the  ecliptic  23°  27 r  47",  the  longi¬ 
tude  of  Rigel  IV  18'  51",  and  the  latitude  31°  8'  45"  S,  to  find 
the  right  ascension  and  declination.  Ans.  Right  Ascen.  76°  28' 
21",  and  declin.  8  25'  1"  S. 

3.  When  the  obliquity  of  the  ecliptic  was  23°  27'  46",  the 


ASTRONOMY. 


289 


longitude  of  Procyon  113°  18'  55",  and  the  latitude  15°  59'  0"  S, 
what  were  the  right  ascension  and  declination?  Ans.  Right  ascen. 
112°  28'  48",  and  declin.  5°  40'  35"  N. 

PROBLEM  XXI. 

The  Obliquity  of  the  Ecliptic,  and  the  Longitude  and  Declination 
of  a  Body  being  given ,  to  find  the  Angle  of  Position. 

Add  together  the  Cosine  of  the  longitude,  the  Sine  of  the  obli¬ 
quity,  and  the  arithmetical  complement  of  the  Cosine  of  the  de¬ 
clination,  taking  them  all  affirmative,  and  reject  10  from  the  in¬ 
dex;  the  result  will  be  the  Sine  of  the  angle  of  position;  which, 
in  all  cases  where  the  problem  is  used  in  calculating  an  occupa¬ 
tion  of  a  planet  or  star,  by  the  moon,  must  be  taken  less  than  90°. 

When  the  longitude  is  less  than  90°  or  more  than  270°,  the 
northern  part  of  the  circle  of  latitude  lies  to  the  west  of  the  circle 
of  declination;  bnt  when  the  longitude  is  between  90c  and  270% 
it  lies  to  the  east. 

Exam.  1.  Given  the  obliquity  of  the  ecliptic  23’  27'  46",  the 
longitude  of  Arcturus  201°  44'  16",  and  the  declination  20°  7' 
5"  N.  to  find  the  angle  of  position. 

cos.  Long.  20P  44'  16"  9,96797 

sin.  Obliq.  23  27  46  9.60005 

cos.  Declin.  20  7  5  Ar.  Co.  0.02734 

sin.  Ang.  Posit.  23  11  46  9.59536 

The  circle  of  latitude  lies  to  the  east  of  the  circle  of  declination. 

2.  Given  the  obliquity  of  the  ecliptic  23"  27'  47",  the  longi¬ 
tude  of  Rigel  74°  18'  51",  and  the  declination  8°  25.'  1"  S;  re¬ 
quired  the  angle  of  position.  Ans.  6°  14'  50". 

3.  When  the  obliquity  of  the  ecliptic  was  23°  27'  46",  the  longi¬ 
tude  of  Procyon  113°  18'  55'',  and  the  declination  5°  40'  35"  N; 
what  was  the  angle  of  position?  Ans.  9°  6'  43  '. 

38 


290 


ASTRONOMY. 


PROBLEM  XXII. 

The  Sutfs  Right  Ascension  on  two  consecutive  days  at  noon,  and 
the  Right  Ascension  of  a  Star  being  given ,  to  find  the  time  of  its 
Passage  over  the  Meridian. 


Subtract  the  sun’s  right  ascension  on  the  first  of  the  two  given 
ilays,  from  that  on  the  second,  and  also  from  the  right  ascension 
of  the  star;  increasing,  when  necessary,  the  latter  quantities  by 
360°,  or  by  24  hours,  according  as  the  right  ascensions  are  ex¬ 
pressed  in  degrees,  or  in  time.  Then,  as  the  first  remainder,  in¬ 
creased  by  360°,  or  by  24  hours  :  the  second  :  :  24  hours  :  the 
time  of  the  star’s  passage  over  the  meridian. 

Note  1.  The  time  of  a  star’s  passage  maybe  found  nearly,  by 
subtracting  the  sun’s  right  ascension  at  noon,  from  the  right  as¬ 
cension  of  the  star,  and  diminishing  the  remainder  by  1,  2,  or  3 
minutes,  according  as  the  remainder  is  near  to  6,  12  or  18  hours. 

2.  The  sun’s  right  ascension  is  given  in  the  Nautical  Almanac, 
for  each  day  at  apparent  noon  on  the  meridian  of  Greenwich,  and 
may  easily  be  found  for  any  other  meridian,  by  proportion. 


Exam.  1.  From  the  Nautical  Almanac,  the  sun’s  right  as¬ 
cension  on  the  11th  of  March,  1821,  at  apparent  noon,  at  Phila¬ 
delphia,  was  23  h.  26  m.  21  sec.  and  on  the  12th,  it  was  23  h. 
30  m.  1  sec.  Required  the  time  at  which  Sirius  passed  the 
meridian,  its  right  ascension  being  then  6  h.  37  m.  16  sec. 

h.  m.  sec.  h.  m.  sec. 

From  23  30  1  From  6  37  16 

Take  23  26  21  Take  23  26  21 


1st.  rem.  3  40 


2d.  rem.  7  10  55 


h.  m.  sec.  h.  m.  sec.  h.  h.  m.  sec. 

24  3  40  :  7  10  55 :  :  24  :  7  9  49  time  required. 

2.  Given  the  sun’s  right  ascension  on  the  10th  of  April,  1821,  at 
apparent  noon,  at  Boston,  1  h.  15  m.  39  sec,,  on  the  11th,  1  h. 
19  m.  19  sec.,  and  the  right  ascension  of  Antares  at  the  same 


ASTRONOMY. 


291 


time  16  h.  18  m.  28  sec.  to  find  the  time  of  its  passage  over 
the  meridian.  Jins.  15  h.  0  m.  31  sec. 

3.  Required  the  time  of  Arcturus'  passage  over  the  meridian 
of  Philadelphia,  on  the  15th  of  August,  1821,  finding  the  right 
ascension  of  the  star  by  Prob.  XIV,  and  the  sun’s  right  ascension 
on  the  15th  and  16th,  from  the  part  of  the  Nautical  Almanac,  con¬ 
tained  in  table  LV.  Jins .  4  h.  27  m.  33  sec. 


PROBLEM  XXIII. 


The  Right  Ascensions  of  the  Sun  and  Moon ,  in  time ,  being 
given ,  on  two  consecutive  days  at  woon,  to  find  the  time  of  the 
Moon's  Passage  over  the  Meridian. 

Subtract  the  right  ascension  of  the  sun  on  the  first  day  at  noon, 
from  that  on  the  second. 

Subtract  the  right  ascension  of  the  moon  on  the  first  day  at 
noon,  from  that  on  the  second. 

Subtract  the  right  ascension  of  the  sun  on  the  first  day  at  noon, 
from  that  of  the  moon,  increasing  the  latter,  when  necessary,  by 
24  hours. 

Add  the  first  remainder  to  24  hours,  and  from  the  sum,  subtract 
tbe  second  remainder.  Then,  as  the  result :  third  remainder  :  : 
24  hours  :  time  of  the  moon’s  passage  over  the  meridian. 

Note.  The  time  of  the  moon’s  passage  over  the  meridian  of 
Greenwich  is  given  for  each  day  in  the  Nautical  Almanac.  The 
times  of  the  passages  of  the  planets  are  also  given  for  several  days 
in  each  month.  See  table  LV. 

Exam.  1.  Given  the  sun’s  right  ascension  on  the  11th  of 
March,  1821,  at  apparent  noon,  at  Philadelphia,  23  h.  26  m. 
21  sec.  and  on  the  12th,  23 h.  30  m.  1  sec.;  the  moon’s  right  as¬ 
cension  on  the  1 1  th,  6  h.  4  m.  35  sec.  and  on  the  12th,  7  h.  2  m. 
47  sec.  required  the  time  of  the  moon’s  passage  over  the  me¬ 
ridian. 


h.  m.  sec. 


h.  m.  sec. 


From  23  30  1 

Take  23  26  21 


From  7  2  47 

Take  6  4  3$ 


1st  remt  3  40 


2d  rem.  58  12 


292 


h.  m.  sec. 
From  6  4  35 
Take  23  26  21 


ASTRONOMY* 


h.  in,  sec. 
From  24  3  40 
Take  58  12 


3drem.  6  38  14  23  5  28 

h.  m.  sec  h.  m.  sec.  h.  h.  m.  sec. 

As  23  5  28  :  6  38  14  :  :  24  :  6  53  54,  time  required. 

2.  Given  the  sun’s  right  ascension  on  the  10th  of  April,  1821, 
at  apparent  noon,  at  Boston,  1  h.  15  m.  39  sec.  and  on  the  11th, 
1  h.  19  m.  19  sec.  the  moon’s  right  ascension  on  the  10th,  8  h. 
35  m.  43  sec.  and  on  the  llth,  9  h.  24  m.  35  sec.  required  the 
time  of  the  moon’s  passage.  Jins.  7  h.  34  m.  20  sec. 

3.  Given  the  sun’s  right  ascension  on  the  13th  of  August,  1821, 
at  apparent  noon  at  Greenwich,  9  h.  30  m.  58  sec.  and  on  the 
14th,  9  h.  34  m.  44  sec.  the  moon’s  right  ascension  on  the  13th, 
21  h.  28  m.  12  sec.  and  on  the  14th,  22  h.  21  m.  36  sec.  required 
the  time  of  the  moon’s  passage.  Ans.  12  h.  22  m.  50  sec. 

PROBLEM  XXIV. 

The  Latitude  of  a  Place  and  the  Sun's  Declination  at  noon 
being  given ,  to  find  the  time  of  his  Rising  and  Setting . 

To  the  Tangent  of  the  latitude  of  the  place,  add  the  Tangent 
of  the  sun’s  declination,  rejecting  10  from  the  index;  the  result 
will  be  the  Sine  of  the  ascensional  difference ,  which  must  be  taken 
less  than  90°,  and  reduced  to  time. 

The  ascensional  difference,  added  to  6  hours,  when  the  latitude 
and  declination  are  both  of  the  same  name,  that  is,  both  north  or 
both  south,  but  subtracted  from  6  hours,  when  they  are  of  differ¬ 
ent  names,  will  give  the  semi-diurnal  arc. 

The  semi-diurnal  arc  expresses  the  time  of  sunset,  and  sub¬ 
tracted  from  12  hours,  gives  the  time  of  sunrise. 

Note.  1.  In  the  above  rule,  no  notice  is  taken  of  the  change 
in  the  sun’s  declination  between  noon  and  the  time  of  his  being- 
in  the  horizon,  nor  of  the  effect  of  refraction  in  changing  the  time 
of  his  rising  and  setting.  When  the  time  of  the  sun’s  apparent 
rising  or  setting  is  required  with  precision,  the  declination  may  be 


ASTRONOMY. 


293 


found  for  the  lime  of  rising  or  setting  as  given  by  the  above  rule, 
and  then  the  calculation,  performed  by  the  formula  in  art.  24, 
chap.  IX.  But  this  is  seldom  necessary. 

2.  The  rising  or  setting  of  a  planet  or  star  may  be  found  by 
calculating  the  semi-diurnal  arc  as  in  the  above  rule,  and  sub¬ 
tracting  it  from  the  time  of  the  body’s  passage  over  the  meridian 
for  the  rising,  and  adding  it,  for  the  setting. 

Exam.  1.  Required  the  time  of  the  sun’s  rising  and  setting  at 
Philadelphia,  on  the  25th  of  January,  1821,  the  declination  at 
noon  of  that  day  being  18'  52'  S. 

tan.  Lat.  39°  57'  -  -  9.92304 

tan.  Decl.  18  52  S  -  -  9.53368 

sin.  Asc.  Diff.  16  38  9.45672 

4 

1  h.  6  m.  32  sec. 

6  0 

Semi-diur.  arc  4  h.  53  m.  time  of  sunset, 

7  h  7  m.  time  of  sunrise. 

2.  Required  the  time  of  the  sun’s  rising  and  setting  at  St. 
Petersburg,  when  the  declination  is  23°  28'  N.  Jins.  Sun  rises 
at  2  h.  46  m.  and  sets  at  9  h.  14  m. 

3.  At  what  time  did  the  sun  rise  and  set  at  Philadelphia,  on  the 
21st  of  August,  1821?  Jins.  Sunrise  5h.  19  m.  and  sunset 
6  h.  41  m. 

PROBLEM  XXV. 

To  reduce  the  time  of  the  Moon's  Passage  over  the  Meridian  of 
Greenwich ,  as  given  in  the  Nautical  Almanacy  to  the  time  of  its 
Passage  over  the  Meridian  of  any  other  Place. 

Take  from  the  Nautical  Almanac,  the  difference  between  the 
time  of  the  moon’s  passage  on  the  given  day,  and  the  next  fol¬ 
lowing  or  next  preceding  day,  according  as  the  place  is  in  west 


294 


ASTRONOMY. 


or  east  longitude.  Then  take  from  table  LVIII,  the  quantity  cor¬ 
responding  to  this  difference  at  the  top  and  the  difference  of  lon¬ 
gitude,  in  time,  at  the  side.  This  quantity  will  be  the  reduction 
and  being  added  to  the  time  of  the  moon’s  passage  over  the  meri¬ 
dian  of  Greenwich  on  the  given  day,  when  the  place  is  in  west 
longitude,  but  subtracted ,  when  it  is  in  east  longitude,  will  give 
the  required  time  of  passage,  in  the  time,  reckoned  at  the  given 
place. 

Exam.  1.  Required  the  time  of  the  moon’s  passage  over  the  me* 
ridian,  at  Philadelphia,  on  the  17th  of  August,  1  @21 . 

li.  m. 

Passage  at  Greenwich,  on  the  17th,  -  15  44 

Reduction  from  table  LVIII.  -  -  -  11 

Passage  at  Philadelphia,  17th.  -  -  15  55 

Or  in  common  reckoning,  on  the  18th  at  3  h.  55  m.  A.  M. 

2.  What  was  the  time  of  the  moon’s  passage  over  the  meri¬ 
dian  of  Boston,  on  the  10th  of  August,  1821?  Jins.  9  h. 
50  m.  P.  M. 

3.  Reduce  the  time  of  the  moon’s  passage  over  the  meridian, 
as  given  in  the  Nautical  Almanac  for  the  21st  of  August,  1821,  to 
the  time  of  passage  at  New  York.  Jins.  On  the  22d  day  at  7  h. 
45  m.  A.  M.  in  common  reckoning. 

PROBLEM  XXVI. 

i  .  *  ,  ...  L.Jli  *  nail 

From  the  Moon's  Declination ,  as  given  in  the  Nautical  Almanac , 

for  each  noon  and  midnight ,  to  find  the  Declination ,  nearly,  for  a 
given  Time  and  Place. 

Reduce  the  given  time  to  apparent  time  at  Greenwich.  Then, 
taking  the  change  in  the  moon’s  declination, for  the  12  hours  within 
which  the  time  at  Greenwich  falls,  find  in  table  LIX,  the  quan¬ 
tities  corresponding  to  the  time  at  the  side,  and  to  the  degrees, 
tens  of  minutes  and  minutes  of  the  change  in  declination,  at  the 
top.  The  sum  of  these,  added  to  the  declination  at  the  noon  or 
midnight  next  preceding  the  time,  when  the  declination  is  in- 


ASTRONOMY.  £9  5 

treasing,  but  subtracted  when  it  is  decreasing,  will  give  the  re¬ 
quired  declination. 

Note  1.  When  the  declinations  at  one  noon  or  midnight,  and 
at  the  following  midnight  or  noon,  are  of  different  names,  their 
sum  is  the  change  in  declination  for  12  hours. 

2.  When  the  sum  of  the  quantities  taken  from  the  table  is  to  be 
subtracted  from  the  declination  and  is  greater  than  it,  the  latter 
must  be  subtracted  from  the  former,  and  the  name  changed  from 
North  to  South,  or  from  South  to  North. 

Exam.  1.  Required  the  moon’s  declination  on  the  15th  of 
August,  1821,  at  10  h.  25  m.  P.  M.  apparent  time  at  Phila¬ 
delphia. 


d.  h.  m. 


Time  at  Philadelphia,  August, 

Diff.  of  long.  - 

15 

10  25 

5  1 

Time  at  Greenwich, 

15 

15  26 

Declination,  the  15th  at  midnight, 
do.  16th  at  noon, 

0°  14'  S 
3  14  N 

Change  in  12  hours, 

- 

3  28 

Declination,  the  15th,  at  midn. 
Sum  of  quantities  from  table  LIX 

- 

0°  14'  S 
0  59 

Required  declin. 

—  *• 

0  45  N. 

2.  Required  the  moon’s  declination,  on  the  18th  of  August, 
1821,  at  4  h.  10  m.  P.  M.  apparent  time  at  Philadelphia.  Jins. 
18°  15'  N. 

3.  Required  the  moon’s  declination,  on  the  2d  of  August, 
1821,  at  1  b.  28  m.  A.  M.  apparent  time  at  New  York.  Jins. 
0°  59'  S. 


296 


ASTRONOMY. 


PROBLEM  XXVII. 

To  find  the  time  of  the  moon’s  Rising  or  Setting  at  a  given 
Place ,  on  a  given  astronomical  day ,  by  the  aid  of  the  Nautical 
Almanac. 

Find  the  time  of  the  moon’s  passage  over  the  meridian  of  the 
given  place  by  Prob.  XXV. 

To,  or  from  the  time  of  the  passage,  according  as  the  moon’s 
setting  or  rising  is  required,  add  or  subtract  6  hours,  and  find,  by 
the  last  problem,  the  moon’s  declination  for  the  resulting  time, 
reduced  to  the  meridian  of  Greenwich. 

With  the  latitude  of  the  place  and  the  moon’s  declination,  find 
the  semidiurnal  arc,  as  in  Prob.  XXIV,  and  apply  it  to  the  time  of 
the  moon’s  passage  over  the  meridian,  by  subtracting  for  the  ri¬ 
sing,  or  adding  for  the  setting;  the  result  will  be  the  approximate 
time  of  rising  or  setting. 

Find  the  moon’s  declination  for  the  approximate  time  of  rising 
or  setting,  reduced  to  the  meridian  of  Greenwich,  and  with  this 
declination,  again  calculate  the  semidiurnal  arc. 

Take  the  difference  between  the  times  of  the  moon’s  passage 
over  the  meridian  of  Greenwich,  on  the  given  day  and  the  next 
preceding,  or  next  following  one,  according  as  the  rising  or  setting 
is  required.  From  table  LVIII,  take  the  quantity  corresponding 
to  this  difference  at  the  top,  and  the  semi-diurnal  arc,  last  found, 
at  the  side.  This  quantity  will  be  a  correction,  which,  added  to 
the  semi-diurnal,  will  give  the  corrected  semi-diurnal  arc. 

Apply  the  corrected  semi-diurnal  arc  to  the  time  of  the  passage 
over  the  meridian  of  the  given  place,  by  subtracting  for  the  moon’s 
rising,  or  adding  for  the  setting;  the  result  will  be  the  required 
time,  sufficiently  accurate  for  all  common  purposes. 

Note.  When  it  is  required  to  make  many  calculations  of  the 
moon’s  rising  or  setting,  for  any  particular  place,  they  may  be 
much  abbreviated  by  little  expedients,  which  it  would  be  trouble¬ 
some  to  specify.  It  may  however  be  observed  that  the  operation 
is  considerably  facilitated  by  having  a  table  of  semi-diurnal  arcs, 
calculated  for  the  latitude  of  the  place,  similar  to  table  LXIV, 
which  is  adapted  to  the  latitude  of  Philadelphia. 


ASTRONOMY.  #  297 


Exam.  1 .  Required  the  time  of  the  moon’s 

rising  at  Philadel* 

liaon  the  18th  of  August,  1821. 

d.  h.  m. 

Passage  over  mer.  of  Greenwich, 

«• 

18  16  37 

Reduction,  - 

12 

Passage  over  mer.  of  Philadelphia, 

. 

18  16  49 

Subtract  - 

6  0 

18  10  49 

Diff.  of  Long.  - 

- 

5  1 

Time  at  Greenwich,  #  -  ■* 

- 

18  15  50 

Moon’s  declin.  on  the  18th,  at  15h. 

50m. 

is  19”  40'  S. 

tan.  Lat.  -  39°  57' 

. 

9.92304 

tan.  Declin.  -  19  40  N. 

- 

9.55315 

sin.  Ascen.  diff.  17  25 

9.47619 

4 

lh.  10m. 

6  0 

Semi-diur.  arc,  7h.  10m. 

d.  h.  m. 

Moon’s  passage  over  mer.  of  Philadelphia, 

,  18  16  49 

Semi-diur.  arc.  -  0 

- 

7  10 

Approximate  time  of  moon’s  rising, 

• 

18  9  39 

Diff.  of  Long.  - 

- 

5  1 

Time  at  Greenwich, 

- 

18  14  40 

Moon’s  declin.  on  the  18th,  at  14h.  40m. 

is  19°  26'  N, 

39 

298 


ASTRONOMY. 


9.92304 

9.54754 


tan.  Lat.  -  39°  57' 

tan.  Declin.  -  19  26  N. 


sin.  Ascen.  diff.  17  11 
4 


9.47058 


lh.  9  m, 
6  0 


Semi-diur.  arc,  7  9 

Correction,  15 


Semi-d.  arc  cor.  7  24 


Moon’s  passage, 
Corrected  Semi-diur.  arc. 


d.  h.  m. 
18  16  49 


7  24 


Time  of  moon’s  rising,  -  -  -  IS  9  25 

2.  Required  the  time  of  the  moon’s  setting,  at  Philadelphia,  on 
the  11th  of  August,  1821.  Jins .  15  h.  38  m.,  or  in  common  reck¬ 
oning,  on  the  12th,  at  3  h.  38  m.  A.  M. 

3.  Required  the  time  of  the  moon’s  rising,  at  New  York,  on 
the  21st  of  August,  1821.  Jins.  11  h.  36  m.  P.  M. 

PROBLEM  XXVIII. 

To  find  the  Longitude  and  Altitude  of  the  JVonagesimal  Degree 
of  the  Ecliptic ,  for  a  given  time  and  place. 

Find  the  reduced  latitude  of  the  place  by  problem  XIII:  and 
when  it  is  north,  subtract  it  from  90°,  but  when  it  is  south,  add  it  to 
90°,  for  the  reduced  distance  of  the  place  from  the  north  pole. 
Take  half  the  difference  between  this  quantity  and  the  obliquity  of 
the  ecliptic:  also,  half  the  sum  of  the  same  quantities.  From  the 
Cosine  of  the  half  difference,  subtract  the  Cosine  of  the  half  sum, 
and  call  the  result,  logarithm  A.  From  the  Tangent  of  the  half 
difference,  with  the  index  increased  by  10,  subtract  the  Tangent 
of  the  half  sum,  and  call  the  result,  logarithm  B..  Also,  call  the 
Tangent  of  the  half  sum,  logarithm  C. 


ASTRONOMY. 


299 


For  the  given  time,  reduced  to  mean  time  at  Greenwich, 
find  the  sun’s  mean  longitude  and  the  argument  N,  from  tables 
IX,  X,  XIII,  and  XIV.  To  the  sun’s  mean  longitude,  increased 
bj  2°,  apply,  according  to  its  sign,  the  nutation  in  right  ascension, 
taken  from  table  XVIII,  with  argument  N,  and  it  will  give  the 
sun’s  mean  longitude,  reckoned  from  the  true  equinox. 

To  the  sun’s  mean  longitude  from  the  true  equinox,  add  the 
mean  time  of  day,  at  the  given  place,  expressed  astronomically 
and  reduced  to  degrees,  and  reject  360°  from  the  sum,  when  it 
exceeds  that  quantity.  The  result  will  be  the  right  ascension  of 
the  mid-heaven .* 

From  the  right  ascension  of  the  mid-heaven,  subtract  90°,  the 
former  being  first  increased  by  360°,  when  necessary,  and  call 
half  the  remainder  R. 

To  the  logarithm  A,  add  the  Tangent  of  R,  and  the  result  will 
be  the  Tangent  of  an  arc  E,  which  must  be  taken  according  to 
the  sign,  but  less  than  180°.  To  the  Tangent  of  E,  add  the  loga¬ 
rithm  of  B,  rejecting  10  from  the  index,  and  the  result  will  be 
the  Tangent  of  an  arc  F,  which  must  also  be  taken  according  to 
the  sign,  and  less  than  180°.  The  sum  of  the  arcs  E  and  F,  and 
90°,  rejecting  360°,  when  the  sum  exceeds  that  quantity,  will  be 
the  longitude  of  the  nonagesimal  degree. 

Add  together  the  logarithm  C,  the  Cosine  of  E,  and  the  arith¬ 
metical  complement  of  the  Cosine  of  F,  and  reject  10  from  the 
index:  the  result  will  be  the  Tangent  of  half  the  altitude  of  the 
nonagesimal  degree. 

Note  1.  The  above  rule,  which  differs  but  little  in  substance 
from  that  given  by  Bowditch  in  his  Practical  Navigator,  is  gene¬ 
ral  for  all  places,  except  within  the  North  polar  circle.  And  the 
only  difference  there,  is,  that  for  the  longitude  of  the  nonagesimal 

*  When  the  sun’s  true  longitude  has  been  previously  calculated  for  the 
same  time,  for  which  the  right  ascension  of  the  mid-heaven  is  wanted,  it  is 
evident  the  tabular  mean  longitude  and  the  argument  N,  are  already 
known. 

It  may  also  be  observed,  that  the  right  ascension  of  the  mid-heaven  is 
equal  to  the  sum  of  the  sun’s  true  right  ascension,  and  the  apparent  time  ex¬ 
pressed  astronomically  and  reduced  to  degrees:  360°  being  rejected  when 
the  sum  exceeds  that  quantity. 


300 


ASTRONOMY. 


degree,  90°  must  be  added  to  the  arc  E,  and  the  arc  F  subtracted 
from  the  sum. 

2.  When  the  longitude  and  altitude  of  the  nonagesimal  degree 
are  required,  at  any  given  place  for  several  different  times  in  the 
same  day,  which  is  generally  the  case,  the  same  logarithms,  A, 
B  and  C,  when  they  have  been  once  found,  will  answer  for  all 
the  other  operations.  Indeed,  the  obliquity  of  the  ecliptic  changes 
so  slowly,  that  except  great  accuracy  is  required,  the  same  loga¬ 
rithms  may  be  used  in  calculations,  for  a  time  several  years  dis¬ 
tant  from  the  time  for  which  they  were  obtained. 

3.  The  last  part  of  the  above  rule  gives  the  distance  of  the 
zenith  of  the  place  from  the  north  pole  of  the  ecliptic,  which  is 
not  always  the  real  altitude  of  the  nonagesimal.  Generally  in  the 
southern  hemisphere,  and  frequently  in  the  northern  hemisphere, 
near  the  equator,  it  is  the  supplement  of  the  altitude.  But  it  sim¬ 
plifies  the  rule  for  the  parallaxes,  to  which  this  problem  is  preli¬ 
minary,  and  produces  no  error,  to  use  the  same  term  in  all  cases. 

Exam.  1.  Required  the  longitude  and  altitude  of  the  nonagesi¬ 
mal  degree  of  the  ecliptic,  at  Philadelphia,  on  the  27th  of  August, 
at  7  h.  30  m.  21  sec.  A.  M.  mean  time,  the  obliquity  of  the  eclip¬ 
tic  being  then  23°  27'  55". 

The  reduced  latitude  of  Philadelphia,  found  by  problem  XIII, 
is  39°  45'  43"  N,  and  this  taken  from  90%  leaves  the  polar  dis¬ 
tance  503  14'  17";  the  difference  and  sum  of  this  quantity  and  the 
obliquity  of  the  ecliptic  are  26°  46'  22"  and  73°  42'  12";  half 
difference  13°  23'  11";  half  sum  36°  51'  6". 

4  diff.  13°  23'  11"  cos.  9.98803  tan.  +  10,  19.37654 
#  sum  36  51  6  cos.  9.90319  tan.  C.  9.87478 

A.  0.08484  B.  9.50176 

The  sun’s  longitude  taken  from  the  tables,  for  the  given  time, 
and  increased  by  2°,  is  5s  5°  24'  38",  and  the  argument  N  is  71. 
The  nutation,  taken  from  table  XVIII,  with  argument  N,  is  +  7". 
Hence,  the  sun  s  mean  longitude  from  the  true  equinox  is  5s  5° 
24' 45",  or  155°  24'  45".  The  given  time  of  day  expressed 


ASTRONOMY. 


301 


astronomically, is  19  h.  30  m.  21  sec.;  which,  in  degrees,  is  292° 
35'  15". 

Given  time,  in  degrees,  -  292°  35'  15" 

Sun’s  mean  long.  -  -  155  24  45 

Right  ascen.  mid-heaven,  -  88  0  0 

90  0  0 


2)358  0  0 
R.  179  0  0 

A.  0.08484 

R  179°  0'  0"  tan.  —  8.24192 

E  178°  47'  4"  tan.  — 8.32676 

B.  9.50176 

F  179  36  50  tan.  — 7.82852 

90  0  0 

- -  3  alt.  non. 

88  23  54  long,  nonages. 


cos.  —  9,99990 
C.  9.87478 

Ar.  Co.  cos. — 0.00001 
36°  50' 47"  tan.  9.87469 
73  41  34  alt.  nonages. 


2.  Required  the  longitude  and  altitude  of  the  nonagesimal  degree 
of  the  ecliptic  at  Philadelphia,  on  the  27th  of  August,  1821,  at 
8  h.  53  m.  20  sec.  A.  M.  mean  time.  Ans.  Long.  105°  2'  18", 
and  alt.  72°  43'  32". 

3.  Required  the  longitude  and  altitude  of  the  nonagesimal  de¬ 
gree,  at  Philadelphia,  on  the  27th  of  August,  1821,  at  10  h.  14  m. 
A.  M.  apparent  time.  Ans.  Long.  121°  21'  25",  and  alt.  69° 
30'  44". 


PROBLEM  XXIX. 

The  Longitude  and  Altitude  of  the  Nonagesimal  Degree  of  the 
Ecliptic ,  and  the  Moon's  True  Longitude ,  Latitude ,  Equatorial 
Parallax ,  and  Horizontal  Semidiameter  being  given ,  to  find  the 
Apparent  Longitude  and  Latitude  as  affected  by  Parallax ,  and  the 
Augmented  Semidiametcr  of  the  Moon,  for  a  given  place . 


302 


ASTRONOMY. 


Find  the  reduction  of  parallax,  by  problem  XIII,  and  subtract 
it  from  the  equatorial  parallax;  and  in  eclipses  of  the  sun ,  subtract 
from  the  remainder,  the  sun’s  parallax,  which  is  8". 7,  or  9"  may 
be  used  without  material  error.  Call  the  last  remainder  the 
Reduced  parallax.  In  occultations  of  a  fixed  star ,  the  first  remain¬ 
der  is  the  reduced  parallax. 

Take  the  difference  between  the  moon’s  longitude  and  the 
longitude  of  the  nonagesimal  degree,  and  call  it  D.  When  the 
moon’s  latitude  is  north ,  subtract  it  from  90°,  but  when  it  is  south , 
add  it  to  90°;  the  difference  or  sum  will  be  the  moon’s  distance 
from  the  north  pole  of  the  ecliptic,  which  call  d .  Call  the  alti* 
tude  of  the  nonagesimal  hy  and  the  reduced  parallax  P. 

Of  the  two  following  methods  of  finding  the  apparent  longitude 
and  latitude,  it  may  be  observed,  that  the  first  is  general,  and  may 
be  used  either  in  eclipses  or  occultations.  The  second  is  applica¬ 
ble,  only  in  eclipses  of  the  sun ,  or  when  it  is  known  that  the  appa¬ 
rent  latitude  is  small.  It  is  more  concise  than  the  first,  and  though 
not  quite  so  accurate,  yet  the  errors  will  seldom  exceed  2  or  3 
tenths  of  a  second.  In  working  by  either  method,  the  student  must 
observe,  that  when  logarithms  are  directed  to  be  added  together, 
the  tens  in  the  resulting  index  are  to  be  rejected.  When  the  loga¬ 
rithm  of  an  arc  is  to  be  taken,  the  arc  must  first  be  reduced  to 
seconds;  and  when  an  arc  is  found,  corresponding  to  a  logarithm, 
it  is  seconds. 

FIRST  METHOD, 

Which  may  be  used ,  either  in  Eclipses  of  the  Sun ,  or  in  Occulta¬ 
tions, 

Add  together  the  logarithm  of  P,  the  Sine  of  h,  and  the  arith¬ 
metical  complement  of  the  Sine  of  d ,  and  call  the  resulting  loga¬ 
rithm  c.  To  the  logarithm  c,  add  the  Sine  of  D,  and  the  result 
will  be  the  logarithm  of  an  arc  u.  Add  together  the  logarithm  c, 
and  the  Sine  of  (D  +  w),  and  the  result  will  be  the  logarithm  of 
an  arc  u'.  Add  together  the  logarithm  c,  and  the  Sine  of  (D  -f  w'), 
and  the  result  will  be  the  logarithm  ofp,  the  parallax  in  longitude. 


ASTRONOMY.  303 

Except  when  great  accuracy  is  required,  the  last  operation  need 
not  be  performed,  and  p  may  be  placed  instead  of  it'. 

Add  p  to  the  moon’s  true  longitude,  when  the  latter  is  greater 
than  the  longitude  of  the  nonagesimal,  but  subtract ,  when  it  is 
less ,  and  the  result  will  be  the  apparent  longitude. 

When  the  apparent  latitude  is  necessarily  small,  as  in  eclipses 
of  the  sun ,  add  together  the  logarithm  of  P,  and  the  Cosine  of  h, 
and  the  result  will  be  the  logarithm  of  an  arc  x.  But  in  occulta- 
tions ,  add  together  the  logarithm  of  P,  the  Cosine  of  /i,  and  the 
Sine  of  d,  and  the  result  will  be  the  logarithm  of  an  arc  v.  To  d, 
add  v,  attending  to  the  sign  of  the  latter.  Then  add  together  the 
logarithm  of  v ,  the  Sine  of  (d  +  ®)j  and  the  arithmetical  comple¬ 
ment  of  the  Sine  of  d,  and  the  result  will  be  the  logarithm  of  the 
arc  x. 

To  d,  add  a?,  attending  to  the  sign  of  the  latter.  Then  add  to¬ 
gether,  the  logarithm  of  P,  marked  negative,  the  Sine  of  h,  the 
Cosine  of  (D  -f  Ip),  and  the  Cosine  of  (d  4-  #),  and  the  result 
will  be  the  logarithm  of  an  arc  z.  The  arc  z,  applied  according  to 
its  sign,  to  the  sum  of  d  and  x ,  will  give  the  apparent  polar  dis¬ 
tance.  And  the  difference  between  this  and  90°,  w  ill  be  the  ap- 
parent  latitude ,  which  will  be  north  or  south ,  according  as  the 
polar  distance  is  less  or  greater ,  than  90°. 

The  sum  of  x  and  z,  regard  being  had  to  their  signs,  will  be 
the  parallax  in  latitude. 

Add  together  the  logarithm  of  the  moon’s  horizontal  semi- 
diameter,  the  Sine  of  the  apparent  polar  distance,  the  Sine  of 
(D  +  w),  the  arithmetical  complement  of  the  Sine  of  d,  and  the 
arithmetical  complement  of  the  Sine  of  D,  and  the  result  will  be 
the  logarithm  of  the  augmented  semidiameter. 

SECOND  METHOD, 

Which  can  only  be  used  when  the  Apparent  Latitude  is  small ,  as  in 
Eclipses  of  the  Sun. 

Add  together,  the  logarithm  of  P,  the  Cosine  of  and  the 
•arithmetical  complement  of  the  Sine  of  d,  and  the  result  will  be 


ASTRONOMY. 


304 

the  logarithm  of  an  arc  x.  Add  together,  the  logarithm  of  x ,  the 
Tangent  of  h,  and  the  Sine  of  D,  and  the  result  will  be  the  loga¬ 
rithm  of  an  arc  u.  Add  together,  the  logarithm  of  «,  the  Sine  of 
(D  -f  w,)  and  the  arithmetical  complement  of  the  Sine  of  D,  and 
the  result  will  be  the  logarithm  of  p ,  the  parallax  in  longitude. 
Take  the  sum  of  d  and  x,  attending  to  the  sign  of  the  latter. 
Then,  add  together,  the  logarithm  of  p ,  the  logarithm  of  the 
difference  between  (d  -f  x )  and  90°,  the  arithmetical  comple¬ 
ment  of  the  logarithm  of  w,  and  the  arithmetical  complement  of 
the  Sine  of  d,  and  the  result  will  be  the  logarithm  of  the  apparent 
latitude,  which  will  be  north  or  south ,  according  as  (d  -f-  x),  is 
less  or  greater  than  90*. 

The  parallax  in  longitude,  p ,  added  to  the  moon’s  true  longi¬ 
tude,  when  the  latter  is  greater  than  the  longitude  of  the  nonagesi- 
mal,  but  subtracted ,  when  it  is  less,  gives  the  apparent  longitude. 

Add  together,  the  logarithm  of  the  moon’s  horizontal  semi¬ 
diameter,  the  Sine  of  (D  +  u ),  the  arithmetical  complement  of 
the  Sine  of  d,  and  the  arithmetical  complement  of  the  Sine  of  D, 
and  the  result  will  be  the  logarithm  of  the  augmented  semi¬ 
diameter.* 

Note.  In  eclipses  of  the  sun,  it  is  not  strictly  the  apparent 
longitude  and  latitude  of  the  moon  that  are  found  by  the  preceding 
rules,  but  the  values  of  those  quantities,  including  the  sun’s  pa¬ 
rallax  in  longitude  and  latitude,  which  are  the  values  wanted  in 
the  calculation. 

Exam.  1 .  About  the  time  of  beginning  of  the  eclipse  of  the 
sun,  on  the  27th  of  August,  1821,  the  longitude  of  the  nonagesi- 
mal  degree,  at  Philadelphia,  was  88°  23'  54",  the  altitude  73°  41 
34",  moon’s  true  longitude  152°  31'  1",  true  latitude  0°  11'  38' 
N,  equatorial  parallax  55'  18",  and  semidiameter  15'  3";  re¬ 
quired  the  apparent  longitude  and  latitude,  and  the  augmented 
semi  diameter. 

*  The  rules  in  the  first  method,  are  deduced  from  formulae  C,  G,  and  Lj 
of  articles  54,  56,  and  57,  chap.  X.  Those  in  the  second,  from  C,  F,  and  L, 
of  articles  54,  55,  and  57. 


ASTRONOMY. 


305 


\ 


Equat.  par. 
Reduction, 

55°  18' 

5 

Moon’s  long.  152°  31'  1' 
Long,  nonag.  88  23  54 

55  13 

D  =  64  7  7 

Sun’s  paral. 

9 

h  =  73  41  34 

P 

=  55  4 

d  =  89  48  22 

BY  THE  FIRST  METHOD. 

P 

3304" 

log.  3.51904 

h  - 

73°  41' 34" 

-  sin.  9.98217 

d 

-  89  48  22 

-  Ar.  Co.  sin.  0.00000 

c.  3.50121 

D  - 

64  7  7 

sin  9,95410 

u  * 

2853" 

-  log.  3.45531 

c.  3.50121 

D  4 -u 

64  54  40 

-  sin.  9.956.96 

u' 

2872" 

log.  3.45817 

c.  3.50121 

D  -f  ti' 

64  54  59 

-  sin.  9  95698 

V 

True  long. 

47'52".l  -  log.  3.45819 

152  31  1 

App.  long. 

153  18  53.1 

P 

- 

-  log.  3.51904 

h  - 

73°  41'  34" 

-  -  cos.  9.44838 

a? 

15  27.7 

-  log.  2.96742 

P  * 

- 

-  log.  — 3.51904 

k 

D  +  ip  - 
d  4-  x 

64  31  3 
90  3  49.7 

-  sin.  9.98217 
cos.  9.63371 
cos.  —  7.04434 

z 

1.5 

-  log.  0.17926 

40 


30(5 


ASTRONOMY. 


Ap.pol.dist.  90°  3'51".2 
90  0  0 


Ap.  lat.  -  0  3  51.2  S. 

Hor.  semidiam.  -  903''  -  -  log.  2.95569 

App.  pol.  dist.  90°  3' 51"  -  -  sin.  10.00000 

D  m  -  -  -  -  -  sin.  9.95696 

d  -  -  -  -  -  Ar.  Co.  sin.  0.00000 

D  Ar.  Co.  sin.  0.04590 


Augmented  semidiam.  15' 9"  -  log.  2.95855 


BY  THE  SECOND  METHOD. 


p 

3304"  - 

log.  3.51904 

h  - 

73° 41' 34" 

cos.  9.44838 

d 

-  89  48  22  -  Ar.  Co. 

sin.  0.00000 

x  - 

15  27.7  - 

log.  2.96742 

h 

-  73  41  34  - 

tan.  10.53379 

D  - 

64  7  7 

sin.  9.95410 

u 

47  33 

log.  3.45531 

D  - 

-  Ar.  Co. 

sin.  0.04590 

D  +  « 

64  54  40 

sin.  9.95696 

P 

47  52  - 

log.  3.45817 

(d  +  x)  — 

90%  3  49.7 

log.  2.36116 

u  - 

Ar.  Co. 

log.  6.54469 

d 

Ar.  Co. 

sin.  0.00000 

App.  lat. 

3  51.2  S. 

log.  2.36402 

Moon’s  true  long.  -  -  152°  31 

'  1" 

P  ~ 

47 

52 

App.  long. 

153  18 

53 

ASTRONOMY. 


307 

Hor.  semidiam.  -  903"  -  -  log.  2.95569 

D-f-tt  -  -  -  -  -  sin.  9.95696 

d  -  -  -  *!  Ar.  Co.  sin.  0.00000 

D  Ar.  Co.  sin.  0.04590 


Augm.  semid.  -  15'  9"  -  -  log.  2.95855 

i  , 

2.  Given  the  longitude  of  the  nonagesimal  67°  29'  8",  the  alti¬ 
tude  57°  56'  36",  the  moon’s  true  longitude  3s  18°  27'  35",  lati¬ 
tude  4°  5'  30"  S,  reduced  parallax  61'  1",  and  horizontal  semi¬ 
diameter  16'  40";  to  find  the  moon’s  apparent  longitude,  latitude, 
and  augmented  semidiameter. 

D  =  40°  58'  27",  h  =  57°  56'  36",  d  =  94°  5'  30",  P  =  61' 1". 


p 

h  - 

d 

3661"  - 

-  57° 56' 36" 

-  94  5  30  -  Ar.  Co. 

log.  3.56360 
sin.  9.92815 
sin.  0.00111 

D 

40  58  27 

c.  3.49286 
sin.  9.81672 

u  - 

34  0  - 

log.  3.30958 

D  -j- it 

-  41  32  27 

c.  3.49286 
sin.  9.82161 

v! 

34  23  - 

log.  3.31447 

D  +  ti' 

-  41  32  50  - 

c.  3.49286 
sin.  9.82167 

P 

34  23.1  - 

log.  3.31453 

True  long.  38  18  27  35 


App.  long.  3  19  1  58.1 


308 


ASTRONOMY. 


p 

- 

log.  3.56360 

h  ■  -  57°  56' 

36" 

-  cos.  9.72490 

d  -  -  94  5 

30  - 

sin.  9.99889 

v  -  32 

18 

log.  3.28739 

d  +  v  -  94  37 

48  - 

sin.  9.99858 

d 

- 

Ar.  Co.  sin.  0.00111 

%  -  32 

16.8 

-  log.  3.28708 

P 

. 

log.  —  3.56360 

h 

- 

sin.  9.92815 

D  +  ip  -  41  15  39 

cos.  9.87605 

d  +  x  -  94  37 

46.8 

-  cos.  —  8.90696 

2  -  -  +  3 

8.3 

-  log.  2.27476 

Ap.  pol.  dist.  94  40 

55.1 

90  0 

0 

App.  lat.  4  40 

55.1 

Hor.  semidiam. 

1000" 

-  log.  3.00000 

App.  pol.  dist.  94°  40'  55" 

-  sin.  9.99855 

D  -  u 

- 

sin.  9.82161 

d 

- 

Ar.  Co.  sin.  0.00111 

D  - 

- 

Ar.  Co.  sin.  0.18328 

Augm.  semidiam.  16'  50".8 

log.  3.00455 

3.  About  the  middle  of  the  eclipse  of  the  sun,  on  the  27th  of 
August,  1821,  the  longitude  of  the  nonagesimal,  at  Philadelphia, 
was  105°  2'  18",  the  altitude  72°  43'  32' ,  moon’s  true  longitude 
153  13'  52",  latitude  0  7'  42"  N.  reduced  parallax  55'  12", 
and  semidiameter  15'  3";  required  the  apparent  longitude  and 
latitude,  and  the  augmented  semidiameter.  Ans.  App.  long.  153° 
53'  27";  app.  lat.  0°  8'  44"  S.;  augm.  semidiam.  15'  12". 2. 

4.  About  the  end  of  the  eclipse  of  the  sun,  on  the  27th  of  Au¬ 
gust,  1821,  the  longitude  of  the  nonagesimal,  at  Philadelphia,  was 
121°  21' 25",  altitude  69°  30'  44",  moon’s  true  longitude  153e 


ASTRONOMY. 


309 


56'  15",  latitude  0°  3'  47"  N,  reduced  parslllax  55'  10",  and 
semidiameter  15'  3";  required  the  apparent  longitude  and  lati¬ 
tude,  and  the  augmented  semidiameter.  Jins.  App.  long.  154°  24; 
21";  app.  lat.  0°  15'  40"  S;  augm.  semidiam.  15'  14".4 

PROBLEM  XXX. 

To  find  from  the  Tables ,  the  Time  of  New  or  Full  Moon,  for  a 
given  Year  and  Month. 

For  Mew  Moon . 

Take  from  table  IV.  the  mean  new  moon  in  January,  for  the 
given  year,  and  the  arguments  I,  II,  III,  and  IV.  Take  from 
table  V,  as  many  lunations,  and  the  corresponding  arguments  I, 
II,  III,  and  IV,  as  the  given  month  is  months  past  January,  and 
add  these  quantities  to  the  former,  rejecting  the  ten  thousands  in 
the  first  two  arguments,  and  the  hundreds  in  the  other  two.  Take 
the  number  of  days  corresponding  to  the  given  month,  from  the 
second  or  third  column  of  table  VI,  according  as  the  given  year  is 
a  common  or  a  bissextile  year,  and  subtract  it  from  the  sum,  in  the 
column  of  mean  new  moon;  the  remainder  will  be  the  tabular  time 
of  mean  new  moon,  in  the  given  month.  If  the  number  of  days, 
taken  from  table  YI,  is  greater  than  the  sum  of  the  days  in  the 
column  of  mean  new  moOn,  as  will  sometimes  be  the  case,  one 
lunation  more  than  is  directed  above,  with  the  corresponding  ar¬ 
guments,  must  be  added. 

With  the  arguments  I,  II,  III,  and  IY,  take  the  corresponding 
equations  from  table  VII,  and  add  them  to  the  time  of  mean  new 
moon;  the  sum  will  be  the  Approximate  time  of  new  moon,  ex¬ 
pressed  in  mean  time  at  Greenwich. 

For  the  approximate  time  of  new  moon,  calculate  by  problems 
VII  and  XI,  the  true  longitudes  and  hourly  motions  in  longitude 
of  the  sun  and  moon.  Take  the  difference  between  the  longitudes, 
and  also  between  the  hourly  motions.  Then,  as  the  difference 
between  the  hourly  motions  :  the  difference  between  the  longi¬ 
tudes  :  :  60  minutes  :  the  correction.  The  correction,  added  to 
the  approximate  time  of  new  moon,  when  the  sun’s  longitude  is 
greater  than  the  moon’s,  but  subtracted ,  when  it  is  to,  will  give 


310 


ASTRONOMY. 


the  true  time  of  new  moon,  expressed  in  mean  time  at  Greenwich, 
This  time  may  be  changed  to  apparent  time,  at  any  given  meri¬ 
dian,  by  problems  VI  and  V. 


For  Full  Moon. 

When  the  time  of  mean  new  moon  in  January  of  the  given 
year  is  on,  or  after  the  16th,  subtract  from  it,  and  the  arguments 
I,  II,  III  and  IV,  a  half  lunation,  with  the  corresponding  argu¬ 
ments,  taken  from  table  V,  increasing  when  necessary,  either  or 
both  of  the  first  two  of  the  former  by  10,000,  and  of  the  two  latter 
by  100;  but  add  them,  when  the  time  is  before  the  16th.  The 
result  will  be  the  tabular  time  of  mean  full  moon  in  January,  and 
the  corresponding  arguments.  Proceed  to  find  the  approximate 
time  of  full  moon,  in  the  same  manner  as  directed  for  the  new 
moon.* 

Calculate  the  true  longitudes  and  hourly  motions  in  longitude 
of  the  sun  and  moon,  for  the  approximate  time  of  full  moon. 
Subtract  the  sun’s  longitude  from  the  moon’s,  and  call  the  re¬ 
mainder  R.  Also,  subtract  the  hourly  motion  of  the  sun  from  that 
of  the  moon.  Then,  as  the  difference  of  the  hourly  motions  :  the 
difference  between  R  and  VI  signs  :  :  60  minutes  :  the  correction. 
The  correction,  added  to  the  approximate  time  of  full  moon, 
when  R  is  less  than  VI  signs,  but  subtracted ,  when  it  is  greater , 
will  give  the  true  time  of  full  moon. 

Exam.  1.  Required  the  time  of  New  Moon  in  August,  1821, 
expressed  in  apparent  time  at  Philadelphia. 

*  When  the  half  lunation  and  arguments  are  to  be  added ,  the  addition 
may  be  left  till  the  proper  number  of  lunations,  with  their  corresponding 
arguments,  are  placed  under,  and  thus  make  one  addition  serve. 


ASTRONOMY, 


311 


M.  New  Moon. 

I. 

II. 

III. 

IV. 

d.  h.  m. 

1821, 

2  17  59 

0092 

7859 

80 

78 

8  lun. 

236  5  52 

6468 

5737 

22 

93 

238  23  51 

6560 

3596 

02 

71 

Days, 

212 

August, 

26  23  51 

1. 

0  54 

II. 

2  13 

III. 

9 

IV. 

10 

August, 

27  3  17 

Approximate  time. 

Sun’s  true  long,  found  for  the  approx,  time 

,  is 

5s  3° 

57' 12 

Moon’s  do. 

5  3 

56  43 

Difference,  - 

- 

0  29 

Moon’s  hourly  motion  in  long,  is 

30'  55" 

Sun’s  do. 

- 

2  25 

Difference,  ... 

28  30 

m.  m.  sec. 

As  28'  30"  :  29"  : :  60  :  1  1,  the  correction. 

d. 

h.  m. 

sec. 

Approx,  time  of  new  moon,  August, 

27 

3  17 

0 

Correction,  - 

+  1 

1 

True  time,  in  mean  time  at  Greenwich, 

27 

3  18 

1 

Equation  of  time,  - 

— -  1 

19 

Apparent  time  at  Greenwich, 

27 

3  16 

42 

Diff.  of  Meridians,  - 

5  0 

46 

Apparent  time  at  Philadelphia, 

26 

22  15 

56 

312 


ASTRONOMY 


2.  Required  the  time  of  Full  Moon  in  July,  1823,  expressed 
in  apparent  time  at  Philadelphia. 


M.  New  Moon. 

I. 

II. 

III. 

IV. 

d.  h.  m. 

1823, 

11  0  20 

0304 

5787 

61 

55 

h  lun. 

14  18  22 

404 

5359 

58 

50 

6  lun. 

177  4  24 

4851 

4303 

92 

95 

202  J3  6 

5559 

5449 

11 

0 

Days, 

181 

July, 

21  23  6 

I. 

2  55 

II. 

13  7 

III. 

5 

IV. 

20 

f  |  Mltll  v*1 

July, 

22  15  33 

Approximate  time. 

Moon’s  true  long,  found  for  the  approx,  time,  is  9s  29°  24'  51" 
Sun’s  do.  -  3  29  25  23 


R.  5  29  59  28 
6  0  0  0 


Diff. 


0  32 


Moon’s  hourly  motion  in  long,  is 
Sun’s  do. 


29'  34' 
2  23 


Difference,  - 


27  11 


m.  m.  sec. 

As  27' 11"  :  32"  : :  60  :  1  11,  the  correction. 


ASTRONOMY. 


313 


Approximate  time  of  full  moon,  July, 
Correction,  - 

True  time,  in  mean  time  at  Greenwich, 
Equation,  - 

Apparent  time  at  Greenwich, 

Diff.  of  meridian,  - 

Apparent  time  at  Philadelphia, 


d.  h.  m.  sec. 
22  15  33  0 

+  1  n 


22  15  34  11 
—  6  2 


22  15  28  9 
5  0  46 


22  10  27  23 


3.  Required  the  time  of  New  Moon  in  July,  1821,  expressed 
in  apparent  time  at  Philadelphia.  Ans.  28  d.  9  h.  9  m.  58  sec. 


P.  M. 


4.  Required  the  time  of  Full  Moon  in  July,  1821,  expressed 
#n  apparent  time,  at  Philadelphia.  Ans.  14  d.  11  h.  17  m.  47  sec. 

PROBLEM  XXXI. 

To  find  the  Time  of  New  or  Full  Moon  in  a  given  Month ,  by 
the  Nautical  Almanac . 

The  times  of  new  and  full  moon  are  given  to  the  nearest 
minute,  on  the  first  page  of  each  month,  in  the  Nautical  Almanac. 
To  find  the  time  of  either,  to  seconds,  call  the  hours  and  minutes 
of  the  time  given  in  the  Almanac,  or  their  excess  above  12 
hours,  T. 


For  New  Moon. 

Take  the  two  longitudes  of  the  moon,  for  the  midnight  and  noon, 
or  noon  and  midnight,  next  preceding  the  time  given  in  the  Al¬ 
manac,  and  also  the  two  immediately  following,  and  place  them 
in  order,  one  below  another.  Do  the  same  with  the  sun’s  longi¬ 
tudes  for  the  same  times,  observing  that  the  sun’s  longitude  at 
midnight  is  half  the  sum  of  the  longitudes,  at  the  preceding  and 
following  noons.  Subtract  each  longitude  of  the  sun,  from  the 
corresponding  longitude  of  the  moon,  noting  the  signs;  the  re¬ 
mainders  will  be  the  distances  of  the  moon  from  the  sun  at  those 

44 


314 


ASTRONOMY. 


times.  Subtract  each  of  these  distances  from  the  one  next  fol¬ 
lowing,  and  the  remainder  will  be  the  first  differences.  Call  the 
middle  one  of  these  A.  Subtract  each  first  difference  from  the 
next  following  one,  for  the  second  differences.  Take  the  mean,  or 
which  is  the  same  thing,  the  half  sum  of  the  second  differences, 
and  call  it  B. 

With  B  at  the  top,  and  the  time  T,  at  the  side,  take  from  ta¬ 
ble  LVI,  the  equation  of  second  differences,  and  apply  it  with  the 
same  sign  as  B,  to  the  second  of  the  distances,  taken  affirmative, 
and  call  the  result  D.  Then,  A  :  D  ::  12  hours  :  time  of  new 
moon.  The  time  thus  obtained  will  be  apparent  time  at  Green¬ 
wich,  and  it  may  be  reduced  to  any  other  meridian  by  prob.  V. 

For  Full  Moon. 

Proceed  exactly  as  for  the  new  moon,  except  that  each  of  the 
sun’s  longitudes  must  be  increased  by  VI  signs. 

Note.  The  times  of  the  first  and  third  quarters  may  be  found, 
to  seconds,  in  the  same  manner,  except  that  the  sun’s  longitudes 
must  be  increased  by  III  or  IX  signs,  instead  of  VI. 

Exam.  1.  Required  the  time  of  new  moOn  in  August,  1821, 
by  the  Nautical  Almanac. 


In  this  example  T  is  3  h.  17  m. 


0’s  Long. 

C  >s  Long. 

Distances. 

1st  Diff. 

2d  Diff. 

26th  midn. 
27th  noon 
27th  midn. 
28th  noon 

4s  26°  2'  0 " 
5  2  15  44 

5  8  26  28 

5  14  34  24 

5s  3°  2  O'  25" 
5  3  49  25 

5  4  18  25 

5  4  47  26 

—  7°  18' 25" 

—  1  33  41 
+  483 
4-  9  46  58 

4-5°44'44" 
A.4-5  41  44 
-j-5  38  55 

—3'  0 " 
—  249 

[B. — 2  54 

Second  distance  -  1°  33'  41" 

Equat.  2d  dilf.  -  -  —  17 


D.  1  33  24 

As  5°  41'  44"  :  1°  33'  24"  ::  12h.  :  3h.  16m.  51  se*\, 
time  of  new  moon,  in  apparent  time  at  Greenwich. 


ASTRONOMY. 


315 


2 .  Required  the  time  of  full  moon,  in  August,  1821,  by  the 
Nautical  Almanac.  Jins.  13th  day,  at  9  h.  7  m.  22  sec.  A.  M. 
apparent  time  at  Philadelphia. 

PROBLEM  XXXII. 

To  determine  what  Eclipses  may  be  expected  to  occur  in  any  given 
i/ear,  and  the  Times  nearly ,  at  which  they  will  take  place . 

For  the  Eclipses  of  the  Sun. 

Take,  for  the  given  year,  from  table  IV,  the  time  of  mean  new 
moon  in  January,  the  arguments  and  the  number  N.*  If  the 
number  N  differs  less  than  53,  from  0,  500,  or  1000,  an  eclipse 
of  the  sun  may  be  expected  at  that  new  moon.  If  the  difference  is 
less  than  37,  there  must  be  one.  When  the  difference  is  between 
37  and  53,  there  is  a  doubt,  which  can  only  be  removed  by  cal¬ 
culation. 

If  an  eclipse  may  or  must  occur  in  January,  calculate  the  ap¬ 
proximate  time  of  new  moon  by  problem  XXX,  and  it  will  be  the 
time  nearly,  at  which  the  eclipse  will  take  place,  expressed  in 
mean  time  at  Greenwich.  This  time  may  be  reduced  to  the  me¬ 
ridian  of  any  other  place  by  problem  V. 

Look  in  column  N  of  table  V,  and,  excluding  the  number  be¬ 
longing  to  the  half  lunation,  seek  the  first  number  that,  added  to 
the  number  N  of  the  given  year,  will  make  the  sum  come  within 
53,  ofO,  500,  or  1000.  Take  the  corresponding  lunations  and 
arguments,  and  this  number  N,  and  add  them  to  the  similar  quan¬ 
tities  for  the  given  year.  Take  from  the  second  or  third  column  of 
table  VI,  according  as  the  given  year  is  common  or  bissextile ,  the 
number  of  days  next  less  than  the  sum  of  the  days  in  the  column 
of  mean  new  moon,  and  subtract  it  from  the  time  in  that  column; 
the  remainder  will  be  the  tabular  time  of  mean  new  moon  in  the 
month  corresponding  to  the  days,  taken  from  table  VI.  At  this 
new  moon  an  eclipse  of  the  sun  may  be  expected;  and  if  the  sum 
of  the  numbers  N,  differs  less  than  37  from  the  numbers  mention- 


*  The  number  N  in  this  table,  designates  the  sun’s  mean  distance  from  the 
moon’s  ascending  node,  expressed  in  thousandth  parts  of  the  circte. 


316 


ASTRONOMY. 


ed  above,  there  must  be  one.  Find  the  time  nearly,  of  the  eclipse, 
by  calculating  the  approximate  time  of  new  moon  as  directed 
above. 

If  there  are  any  other  numbers  in  the  column  N  of  table  V, 
that  when  added  to  the  number  N  of  the  given  year,  will  make 
the  sum  come  within  the  limit  53,  proceed  in  a  similar  manner  to 
find  the  times  of  the  eclipses. 

Note.  When  the  time  at  which  an  eclipse  of  the  sun  will 
take  place  is  thus  found,  nearly,  and  reduced  to  the  meridian  of 
a  given  place  in  north  latitude,  if  it  comes  during  the  day  time, 
and  if  the  sum  of  the  numbers  N,  or  the  number  N  itself  when 
the  eclipse  is  in  January,  is  a  little  above  0,  or  a  little  less  than 
500,  there  is  a  probability  that  the  eclipse  will  be  visible  at  the 
given  place.  When  the  number  N  in  January,  or  the  sum  of  the 
numbers  N,  in  other  months,  is  more  than  500,  the  eclipse  will 
seldom  be  visible  in  northern  latitudes,  except  near  the  equator. 

For  the  Eclipses  of  the  Mam. 

When  the  time  of  new  moon  in  January  of  the  given  year  is  on,  or 
after  the  16th,  subtract  from  it,  from  the  arguments,  and  the  number 
N,  a  half  lunation,  the  corresponding  arguments,  and  the  number 
N;  blit  when  it  is  before  the  16th,  add  them.  The  results  will 
be  the  time  of  mean  full  moon  in  January,  and  the  corresponding 
arguments,  and  number  N.  Proceed  to  find  the  times  at  which, 
eclipses  of  the  moon,  may  or  must  occur,  exactly  as  directed  for 
the  sun,  except  that  the  limits  35  and  25 ,  must  be  used  instead  of 
53  and  37. 

Note.  In  an  eclipse  of  the  moon,  when  the  time  is  found  nearly, 
and  reduced  to  the  meridian  of  a  given  place,  if  it  comes  in  the 
night,  it  will  be  visible  at  that  place. 

Exam.  1.  Required  the  eclipses  that  may  be  expected  in  the 
year  1822,  and  the  times  nearly,  at  which  they  will  take  place. 


ASTRONOMY, 


SI 


For  the  Eclipses  of  the  Sun. 


M.  New  Moon. 

I. 

II. 

HI. 

IV. 

N  ‘ 

1822, 
1  lun. 

d.  h.  m. 

21  15  32 

29  12  44 

0602 

S08 

7182 

717 

78 

15 

66 

99 

930 

85 

51  4  16 

31 

1410 

7899 

93 

65 

15 

Feb. 

I. 

II. 

III. 

IV. 

20  4  16 

7  38 

19  29 

13 

11 

As  the  sum  of  the  numbers  N, 
comes  within  37  of  0,  there  must  be 
an  eclipse. 

Feb. 

21  7  47 

Mean  time  at  Greenwich. 

M.  New  Moon. 

I. 

II. 

III. 

IV. 

N. 

1822, 
7  lun. 

d.  h.  m. 

21  15  32 
206  17  8 

0602 

5659 

7182 

5020 

78 

7 

66 

94 

930 

596 

228  8  40 
212 

6261 

2202 

85 

60 

526 

August, 

I. 

II. 

III. 

IV. 

16  8  40 

1  24 

0  40 

16 

14 

As  the  sum  of  the  numbers  N,  comes 
within  37  of  500,  there  must  be  an 
eclipse. 

August, 

16  11  14 

Mean  time  at  Greenwich 

318 


ASTRONOMY 


1822 
h  lun. 


Hun. 


M.  New  Moon. 

I. 

II. 

III. 

IV. 

N. 

d.  h.  m. 

21  15  32 

0602 

7182 

78 

66 

930 

14  18  22 

404 

5359 

58 

50 

43 

6  21  10 

0198 

1823 

20 

16 

887 

29  12  44 

808 

717 

15 

99 

85 

36  9  54 

1006 

2540 

35 

15 

972 

31 


Feb. 

5  9  54 

I. 

6  52 

II. 

0  20 

III. 

4 

IV. 

29 

Feb. 

5  *17  39 

As  the  sum  of  the  numbers  N,  although  it 
comes  within  35  of  1000,  does  not  come  with¬ 
in  25,  the  eclipse  may  be  considered  doubtful. 
It  may,  however,  be  observed,  that  further 
calculation  by  the  next  problem  would  show 
that  tl^ere  will  be  a  small  eclipse. 


Mean  time  at  Greenwich. 


M.  Full  Moon. 

I. 

II. 

III. 

IV. 

N. 

d.  h.  m. 

1822 

6  21  10 

0198 

1823 

20 

16 

8S7 

7  lun. 

206  17  8 

5659 

5020 

7 

94 

596 

213  14  18 
212 

5857 

6843 

27 

10 

483 

August, 

I. 

II. 

III. 

IV. 

1  14  18 

2  14 

19  26 

3 

26 

As  the  sum  of  the  numbers  N,  comes 
within  25  of  500,  there  must  be  an 
eclipse. 

August, 

2  12  27 

Mean  time  at  Greenwich. 

2.  Required  the  eclipses  that  may  be  expected  in  1823,  and 
the  times  nearly,  at  which  they  will  take  place,  expressed  in 
mean  time  at  Greenwich.  Ans.  One  of  the  moon  on  the  26th  of 
January,  at  5  h.  24  m.  P.  M.;  one  of  the  sun  on  the  11th  of 
February,  at  3  h.  12  m.  A.  M.;  one  of  the  sun  on  the  8th  of  July, 
at  6  h.  50  m.  A.  M.;  and  one  of  the  moon  on  the  23d  of  July,  at 
3h.  33  m.  A.  M. 


ASTRONOMY. 


319 


PROBLEM  XXXIII. 

To  Calculate  an  Eclipse  of  the  Moon. 

Find  the  approximate  time  of  full  moon,  by  prob.  XXX,  and 
for  this  time,  calculate  the  sun’s  longitude,  semidiameter  and 
hourly  motion,  and  the  moon’s  longitude,  latitude,  equatorial  pa¬ 
rallax,  semidiameter  and  hourly  motions  in  longitude  and  latitude. 
Then  find  ihe  true  time  of  full  moon  as  directed  in  prob.  XXX,. 
and  reduce  it  to  apparent  time  at  the  place  for  which  the  calcula¬ 
tion  is  to  be  made.  Call  the  reduced  time,  T. 

For  the  Moon's  Latitude  at  the  True  Time  of  Full  Moon . 

As  1  hour  :  correction  for  the  time  of  full  moon  :  :  moon’s 
hourly  motion  in  latitude  :  correction  of  latitude.  When  the  true 
time  of  full  moon,  expressed  in  mean  time  at  Greenwich,  is  later 
than  the  approximate  time,  the  correction  of  latitude  must  be 
added ,  if  the  latitude  is  increasing ,*  but  subtracted ,  if  it  is  de¬ 
creasing;  but  when  the  true  time  is  earlier  than  the  approximate 
time,  the  correction  must  be  subtracted ,  if  the  latitude  is  increasing , 
but  added ,  if  it  is  decreasing.  The  result  will  be  the  moon’s  lati¬ 
tude  at  the  true  time  of  full  moon. 

For  the  Semidiameter  of  the  Earth's  Shadow . 

To  the  moon’s  equatorial  parallax,  add  the  sun’s,  which  may 
be  taken  9",  and  from  the  sum,  subtract  the  semidiameter  of  the 
sun.  Increase  the  result  by  a  ^  part,  and  it  will  be  the  semi  • 
diameter  of  the  earth’s  shadow,  which  call  S. 

For  the  Inclination  of  the  Moon's  Relative  Orbit. 

To  the  arithmetical  complement  of  the  logarithm  of  the  differ¬ 
ence  between  the  hourly  motions  in  longitude  of  the  moon  and 
sun,  add  the  logarithm  of  the  moon’s  hourly  motion  in  latitude, 
and  the  result  will  be  the  Tangent  of  the  inclination,  which  call  I. 

*  When  the  moon’s  latitude  is  north,  tending  north,  or  south,  tending 
south,  it  is  increasing-,-  but  when  it  is  north,  tending  south,  or  south,  tending 
north,  it  is  decreasing. 


320 


ASTRONOMY, 


Add  tog-ether  the  constant  logarithm  3.55630,  the  Cosine  of  I, 
and  the  arithmetical  complement  of  the  difference  between  the 
hourly  motions  of  the  moon  and  sun,  in  longitude,  rejecting  the 
fens  in  the  index,  and  call  the  resulting  logarithm  R. 

For  the  Time  of  the  Middle  of  the  Eclipse . 

Add  together  the  logarithm  R,  the  logarithm  of  the  moon’s  lati¬ 
tude  at  the  true  time  of  full  moon,  and  the  Sine  of  I,  rejecting 
the  tens  in  the  index,  and  the  result  will  be  the  logarithm  of  an 
interval  J,  in  seconds  of  time,  which,  added  to  T,  when  the  lati¬ 
tude  is  decreasing ,  but  subtracted ,  when  it  is  increasing ,  will  give 
the  time  of  the  middle  of  the  eclipse. 

For  the  Times  of  Beginning  and  End . 

To  the  logarithm  of  the  moon’s  latitude  at  the  true  time  of  full 
moon,  add  the  Cosine  of  I,  rejecting  the  tens  in  the  index,  and 
the  result  will  be  the  logarithm  of  an  arc,  which  call  c.  Call  the 
moon’s  semidiameter,  d. 

To,  and  from,  the  sum  of  S  and  d,  add  and  subtract  c.  Then 
add  together  the  logarithms  of  the  results,  S  -f  d  4-  c  and  S  +  d 
—  c,  divide  the  sum  by  2,  and  to  the  quotient  add  the  logarithm 
R,  and  the  result  will  be  the  logarithm  of  an  interval  x ,  in  seconds 
of  time,  which  subtracted  from,  and  added  to,  the  time  of  the 
middle,  will  give  the  times  of  the  beginning  and  end. 

Note.  If  c  is  equal  to,  or  greater  than  the  sum  of  S  and  d, 
there  can  not  be  an  eclipse. 

\  ^  *  5 '  1  ‘  t  •  ♦  m  "* 

For  the  Times  of  Beginning  and  End  of  the  Total  Eclipse. 

To,  and  from,  the  difference  of  S  and  d,  add  and  subtract  e. 
Then  add  together  the  logarithms  of  the  results,  S  —  d  +  c  and 
S  —  d  —  c,  divide  the  sum  by  2,  and  to  the  quotient  add  the 
logarithm  R,  and  the  result  will  be  the  logarithm  of  an  interval 
x\  in  seconds  of  time,  which  subtracted  from,  and  added  to,  the 
time  of  the  middle,  will  give  the  times  of  the  beginning  and  end 
of  the  total  eclipse. 

Note.  When  c  is  greater  than  the  difference  of  S  and  d,  the 
eclipse  can  not  be  total. 


ASTRONOMY. 


321 


For  the  Quantity  of  the  Eclipse. 


Add  together  the  constant  logarithm  0.77815,  the  logarithm  of 
(S  -f  d  —  c),  and  the  arithmetical  complement  of  the  logarithm 
of  d,  rejecting  the  tens  in  the  index,  and  the  result  will  be  the 
logarithm  of  the  quantity  of  the  eclipse,  in  digits. 

Note  1.  In  partial  eclipses  of  the  moon,  the  southern  part  of 
the  moon  is  eclipsed  when  the  latitude  is  nortl^md  the  northern 
part  when  the  latitude  is  south. 

2.  When  the  eclipse  commences  before  sunset,  the  moon  rises 
about  the  same  time  the  sun  sets.  To  obtain  the  quantity  of  the 
eclipse  nearly,  at  the  time  the  moon  rises,  take  the  difference  be¬ 
tween  the  time  of  sunset  and  the  middle  of  the  eclipse.  Then,  as 
1  hour  :  this  difference  :  :  difference  between  the  hourly  motions 
of  the  moon  and  sun,  in  longitude  :  a  fourth  term.  Add  together 
the  squares  of  this  fourth  term  and  of  the  arc  c,  both  in  seconds, 
and  extract  the  square  root  of  the  sum.  Use  this  root  instead  of 
c,  in  the  above  rule,  and  it  will  give  the  quantity  of  the  eclipse  at 
the  time  of  the  moon’s  rising,  very  nearly.  When  the  eclipse  ends 
after  sunrise  in  the  morning,  the  quantity  at  the  time  of  the  moon’s 
setting  may  be  found  in  the  same  manner,  only  using  sunrise  in¬ 
stead  of  sunset. 

Exam.  1.  Required  to  calculate,  for  the  meridian  of  Philadel¬ 
phia,  the  eclipse  of  the  moon,  in  July,  1823. 

The  approximate  time  of  full  moon,  is  July  22,  at  15h.  33  m. 


Sun’  longitude  at  that  time. 
Do.  hourly  motion, 

Do.  semidiameter, 

Moon’s  longitude, 

Do.  latitude, 

Do.  equatorial  parallax, 

Do.  semidiameter, 

Do.  hor.  mot.  in  long. 

Do.  do„  in  lat.  - 


9  29  24  51 

9  10  N. 

54  1 

d.  14  43 
29  34 

2  43,  tending  north. 


3s  29°  25'  23" 


2  23 
15  46 


42 


322  ASTRONOMY. 

d.  h.  m.  sec. 

Approx,  time  of  full  moon,  July,  22  15  33  0 

Correct,  found  by  prob.  XXX.  -  -fill 


True  time,  in  mean  time  at  Greenwich,  22  15  34  11 
Equat.  of  time,  -  -  —  6  2 


Apparent  ti^|  at  Greenwich,  -  22  15  28  9 

DifF.  of  Long.  -  -  -  5  0  46 


Apparent  time  at  Philadelphia,  T.  22  10  27  23 
m  m.  sec. 

As  60  :  1  11  : :  2'  43"  :  3",  the  correct,  of  lat. 


Moon’s  lat.  at  approx,  time,  -  9'  10"N. 

Correction,  -  -  -  -f  3 

Moon’s  lat.  at  true  time,  -  -  9  13  N. 

Moon’s  equatorial  parallax,  -  54'  1" 

Sun’s  do.  9 

Sum,  -  -  -  -  54  10 

Sun’s  semidiameter,  «  -  15  46 

38  24 

Add  -  -  -  -  0  38 

Semidiam.  of  earth’s  shadow,  -  S.  39  2 


Moon’s  hor.  mot.  less  sun’s,  1631"  Ar.  Co.  log.  6.78755 
Moon’s  hor.  mot.  in  lat.  163  -  log.  2.21219 


I 


5°  42' 


tan.  8.99974 


ASTRONOMY, 


323 


3.55630 

l  ...  5  42 

cos.  9.99785 

Moon’s  bor.  mot.  less  sun’s, 

Ar.  Co.  log.  6.78755 

R.  0.34170 

Moon’s  lat.  -  -  553 

ft 

log.  2.74272 

I  -  -  -  5°  42' 

- 

sin.  8.99704 

t  121  sec.  =  2  m.  1  sec. 

T  -  10  h.  27  m.  23  sec. 

log.  2.08146 

Middle,  10  h.  25  m.  22  sec. 

Moon’s  lat. 

log.  2.74272 

I 

- 

-  cos.  9.99785 

c  -  550"  =  9'  10" 

- 

log.  2.74057 

S  +  d  +  c  -  3775' 

log.  3.57692 

s  +  d  — c  -  2675 

log.  3.42732 

2)7.00424 

3.50212 

;  * .  ’ 

R.  0.34170 

sec.  li.  m.  sec. 
s  =  6980  =  1  56  20 

log.  3.84382 

Middle, 

h. 

10 

1 

m.  sec. 

25  22 

56  20 

Beginning, 

8  29  2 

End, 

12  21  42  A.  M.  of  23d  day. 

S  —  d+c  -  2009' 

S  —  d  —  c  -  909 

log.  3.30298 
log.  2.95856 

2)6.26154 

3.13077 
R.  0.34170 

sec.  m.  sec. 

a;'  =  2968  =  49  28 

log.  3.47247 

log.  3.47247 


3i24  ASTRONOMY. 


Middle, 
of  - 

h.  m.  sec. 
10  25  22 

49  28 

Beginning  of  the  total  eclipse, 

9  35  54 

End  do. 

- 

11  14  50 

S  4-  d  —  c 
d 

883" 

Ar.  Co. 

log. 

log. 

0.77815 

3.42732 

7.05404 

Digits  eclipsed, 

-  18.2 

- 

log. 

1.25951 

2.  Required  to  calculate  for  the  meridian  of  Philadelphia,  the 
eclipse  of  the  moon,  on  the  2d  of  August,  1822. 

Jins.  Moon  rises  about  sunset,  8§  digits  eclipsed, 

Ecliptic  opposition,  -  -  7h.  lGm. 

Middle,  -  -  -  -  7  23 

End,  -  -  8  55 

Digits  eclipsed  9,  on  moon’s  northern  limb. 

3.  Required  to  calculate  for  the  meridian  of  Philadelphia,  the 
eclipse  of  the  moon,  in  January,  1824. 

h.  m. 


Jins .  Beginning,  January  16th,  at  2  17  A.M. 

Middle,  -  -  -  -  3  43 

Ecliptic  opposition,  -  -  -  3  49 

End, . 5  8 


Digits  eclipsed  9.4,  on  moon’s  northern  limb. 

PROBLEM  XXXIV. 

To  Project  an  Eclipse  of  the  Moon. 

Find  the  true  time  of  full  moon,  the  moon’s  latitude  at  that 
time,  the  semidiameter  of  the  earth’s  shadow,  the  sun’s  hourly 
motion,  and  the  moon’s  semidiameter  and  hourly  motions  in  longi¬ 
tude  and  latitude,  as  directed  in  the  last  problem. 


ASTRONOMY. 


325 

To  the  moon’s  hourly  motion  from  the  sun  in  longitude,*  add 
9",  and  it  will  give  the  moon's  hourly  motion  from  the  sun ,  on  the 
Relative  Orbit ,  with  sufficient  accuracy  for  a  construction. 

Draw  any  right  line  AB,  Fig.  56,  for  a  part  of  the  ecliptic,  and 
in  it,  take  a  point  C.  Take  the  semidiameter  of  the  earth’s 
shadow  from  a  scalef  of  equal  parts,  and  with  the  centre  C,  de* 
scribe  a  circle  RST,  to  represent  a  section  of  the  earth’s  shadow. 
Through  C,  draw  KCL,  perpendicular  to  AB.  Take  the  moon’s 
latitude  from  the  scale,  and  set  it  on  the  line  KL,  from  C  to  M, 
above  AB,  when  the  latitude  is  north ,  but  below ,  when  it  is  south . 
Then  M  will  be  the  moon  's  place  at  the  time  of  ecliptic  opposition, 
or  full  moon.  Draw  Mb  parallel  to  AB,  and  to  the  left  of  KL,  and 
make  it  equal  to  the  moon’s  hourly  motion  from  the  sun  in  longi¬ 
tude.  Make  be  perpendicular  to  Mb ,  and  equal  to  the  moon’s 
hourly  motion  in  latitude,  drawing  it  above  Mb,  when  the  latitude 
is  tending  north ,  but  below ,  when  it  is  tending  south .  Through  M 
and  c,  draw  the  indefinite  right  line  PQ,  and  it  will  be  the  moon’s 
relative  orbit. 

Make  the  proportion,  as  60  minutes  :  minutes  and  seconds  of 
the  true  time  of  full  moon  : :  moon’s  hourly  motion  from  the  sun 
on  the  relative  orbit  :  a  fourth  term.  Take  this  fourth  term  from 
the  scale,  and  lay  it  on  the  relative  orbit  from  M  to  the  right  hand , 
and  it  will  give  the  moon’s  place  at  the  whole  hour  next  preceding 
the  time  of  full  moon.  Place  the  number  of  the  hour  to  the  point 
thus  obtained.  Then  commencing  at  this  point,  and  with  the 
moon’s  hourly  motion  from  the  sun  on  the  relative  orbit,  in  the 
dividers,  set  off  equal  spaces  on  the  orbit,  on  each  side  of  the 
point,  and  thus  obtain  the  moon’s  places  at  some  of  the  whole 
hours,  preceding  and  following  the  hour,  mentioned  above.  Put 
the  numbers  of  the  hours  to  these  places.  Divide  each  hour  space 
into  four  equal  parts,  for  quarters,  and  these  into  five  minute  or 
minute  spaces. 

Through  C,  draw  SCT  perpendicular  to  PQ,  and  the  inter- 

*  Which  is  the  difference  of  their  hourly  motions  in  longitude, 
f  A  suitable  scale  is  one  of  10  minutes  to  an  inch.  It  may  also  be  observed, 
that  it  is  most  convenient,  to  reduce  the  seconds  in  the  quantities  to  be 
taken  off,  to  decimals  of  a  minute. 


8£G 


ASTRONOMY. 


section  F,  will  be  the  moon’s  place  at  the  middle  of  the  eclipse. 
With  the  centre  C,  and  a  radius  equal  to  the  sum  of  the  semi- 
diameters  of  the  earth’s  shadow  and  moon,  describe  arcs  cutting 
PQ  in  D  and  II,  the  moon’s  places  at  the  beginning  and  end  of 
the  eclipse.  With  the  same  centre,  and  a  radius  equal  to  the  dif¬ 
ference  of  the  semidiameters  of  the  earth’s  shadow  and  moon,  pro¬ 
vided  this  difference  is  greater  than  CF,  describe  other  arcs,  cut¬ 
ting  PQ  in  E  and  G,  the  moon’s  places  at  the  beginning  and  end 
of  the  total  eclipse.  If  the  difference  of  the  semidiameters  of  the 
earth’s  shadow  and  moon  is  less  than  CF,  the  eclipse  will  not  be 
total. 

From  the  divisions  on  the  relative  orbit,  the  times  at  which  the 
moon  is  at  the  points  D,  F  and  H,  and  consequently  the  times  of 
the  beginning,  middle  and  end  of  the  eclipse,  are  easily  estimated. 
In  like  manner,  when  the  eclipse  is  total,  the  beginning  and  end 
of  the  total  eclipse  are  determined  from  the  points  E  and  G. 

With  the  moon’s  semidiameter  for  a  radius,  and  the  centres 
D,  F  and  II.  describe  circles  to  represent  the  moon  at  the  begin¬ 
ning,  middle  and  end  of  the  eclipse. 

Take  the  distance  NS,  when  the  latitude  is  north,  but  UT, 
when  the  latitude  is  south,  and  measure  it  on  the  scale.  Then, 
as  the  moon’s  semidiameter  :  this  distance  :  :  6  digits  :  the  digits 
eclipsed. 

Note.  The  quantities  used  in  constructing  an  eclipse  are  fre¬ 
quently  called  the  Elements.  It  is  convenient  to  have  them  col¬ 
lected  in  order,  before  commencing  the  construction.  The  true 
time  of  full  moon,  which  is  one  of  the  elements,  may  be  expressed 
either  astronomically  or  in  common  reckoning;  the  former  is  per¬ 
haps  the  most  convenient. 

Exam.  1.  Required  to  construct  the  eclipse  of  the  moon,  in 
July,  1823,  taking  the  time  for  the  meridian  of  Philadelphia. 

The  elements  for  this  construction,  the  most  of  which  have 
been  found  in  the  1st  example  of  the  last  problem,  are  as  fol¬ 
lows: 


ASTRONOMY. 


m 


Elements  Collected. 

True  time  of  full  moon,  July  22d,  10  h. 

Semidiam  of  earth’s  shadow, 

Moon’s  latitude,  north ,  - 

Moon’s  hor.  mot.  from  sun,  in  long. 

Moon’s  hor.  mot.  in  lat.  tending  north , 

Moon’s  hor.  mot.  from  sun,  on  rel.  orb. 

Moon’s  semidiameter,  - 

Sum  of  semidiam.  of  earth’s  shadow  and  moon, 

Difference  of  *  do. 


27  m.  23  sec, 

39'  2"  =  39'.03 
9  13=  9.22 
27  11  =  27.18 
2  43  =  2.72 
27  20  =  27.33 
14  43  =  14.72 
53  45  =  53.75 
24  19  =  24.32 


Draw  Al},  Fig.  56,  take  the  point  C,  and  through  it  draw  K 
CL,  perpendicular  to  AB.  Take  the  moon’s  latitude  9.22,  from 
the  scale,  and  lay  it  on  KL,  from  C  to  M,  above  AB,  because  the 
latitude  is  north.  Draw  Mb  parallel  to  AB,  and  make  it  equal  to 
27.18,  the  moon’s  hourly  motion  from  the  sun  in  longitude.  Draw 
be  perpendicular  to  M6,  on  the  upper  side,  because  the  latitude  is 
tending  north,  and  make  it  equal  to  2.72,  the  moon’s  hourly  mo¬ 
tion  in  latitude.  Through  M  and  c,  draw  the  relative  orbit,  PQ. 

As  60  minutes  :  27  m.  23  sec.  the  minutes  and  seconds  of  the 
true  time  of  full  moon  :  :  27.33,  the  moon’s  hourly  motion  from 
the  sun  on  the  relative  orbit  :  12.47,  the  fourth  term.  Take  this 
fourth  term  and  lay  it  on  the  relative  orbit,  from  M  to  10,  the 
moon’s  place  at  the  10th  hour,  in  this  example.  Take  27.33,  the 
moon’s  hourly  motion  from  the  sun  on  the  relative  orbit,  and  lay 
it  from  10  to  9,  and  9  to  8,  and  on  the  other  side,  from  10  to  1 1, 
1 1  to  12,  and  12  to  13,  for  the  moon’s  places  at  those  hours.  Di¬ 
vide  the  hour  spaces  into  quarters,  and  these  into  five  minute  or 
minute  spaces. 

Through  C,  draw  SCT  perpendicular  to  PQ,  intersecting  it  in 
P,  which  will  be  the  moon’s  place  at  the  middle  of  the  eclipse. 
With  the  radius  53.75,  the  sum  of  the  semidiameters  of  the 
earth’s  shadow  and  moon,  and  the  centre  C,  describe  arcs,  cut¬ 
ting  the  relative  orbit  in  D  and  H,  the  moon’s  places  at  the  be¬ 
ginning  and  end  of  the  eclipse.  With  the  radius  24.32,  the  dif¬ 
ference  of  the  semidiameters  of  the  earth’s  shadow  and  moon, 
describe  arcs,  cutting  the  relative  orbit  in  E  and  G,  the  moon’s 


328 


ASTRONOMY. 


places  at  the  beginning  and  end  of  the  total  eclipse.  The  times 
designated  by  the  points  D,  F,  Hs  E  and  G,  agree  nearly  with  the 
beginning,  middle  and  end  of  the  eclipse,  and  beginning  and  end 
of  the  total  eclipse,  found  in  the  1st  example  of  the  last  problem. 

With  14.72,  the  moon’s  semidiameter,  for  a  radius,  describe 
the  circles  about  the  centres  D,  F  and  IL  Take  the  distance  NS, 
and  measure  it  on  the  scale,  and  it  will  be  found  to  be  about 
44.65.  Then,  14.72  :  44.65  :  :  6  digits  :  18.2  digits,  the  quan¬ 
tity  of  the  eclipse. 

2.  Construct  the  eclipse  of  the  moon,  mentioned  in  the  2d  ex¬ 
ample  of  the  last  problem,  and  the  results  will  be  found  to  agree 
nearly  with  the  answer  there  given. 

3.  Construct  the  eclipse  of  the  moon,  mentioned  in  the  3d  ex¬ 
ample  of  the  last  problem. 

PROBLEM  XXXV. 

To  Project  an  Eclipse  of  the  Sun ,  for  a  given  place. 

Calculate  the  approximate  time  of  new  moon  by  prob.  XXX, 
and  for  that  time,  calculate  the  sun’s  longitude,  semidiameter  and 
hourly  motion,  and  the  moon’s  longitude,  latitude,  equatorial  pa¬ 
rallax,  semidiameter  and  hourly  motions  in  longitude  and  latitude. 
Find  the  true  time  of  new  moon  by  prob.  XXX,  and  reduce  it  to 
apparent  time  at  the  given  place,  expressing  it  astronomically. 
Also,  find  the  moon’s  latitude  at  the  true  time  of  new  moon,  from 
the  hourly  motion  in  latitude,  in  the  same  manner  as  directed  in 
prob.  XXXIII,  for  finding  the  latitude  at  the  true  time  of  full 
moon.  With  the  sun’s  longitude  at  the  approximate  time  of  new 
moon,  neglecting  the  seconds,  and  taking  the  obliquity  of  the 
ecliptic  23°  28',  find  the  sun’s  declination  by  prob.  VII.  Find 
the  moon’s  hourly  motion  from  the  sun  on  the  relative  orbit,  by 
adding  9"  to  the  difference  of  their  hourly  motions  in  longitude. 
Find  the  reduced  latitude  of  the  place  and  the  reduced  parallax, 
by  prob.  XIII.  From  the  moon’s  reduced  parallax,  subtract  the 
sun’s  parallax,  which  may  be  taken  9",  and  the  remainder  will 
be  the  Semidiameter  of  the  Circle  of  Projection. 


ASTRONOMY. 


329 


Draw  a  right  line  AB,  Fig.  57,  and  in  it  take  a  point  C.  Take 
the  semidiameter  of  the  circle  of  projection  from  a  scale  of  equal 
parts,  and  with  the  centre  C,  describe,  on  the  upper  side  of  AB, 
the  semicircle  ADB,  to  represent  the  northern  half  of  the  circle 
of  projection.  When  the  latitude  of  the  place  is  south ,  the  whole 
circle  must  be  described.  Through  C,  and  perpendicular  to  AB, 
draw  the  line  TCY,  to  represent  the  universal  meridian.  With  a 
sector,*  opened  to  the  radius  AC  or  CB,  set  off  from  D,  the  arcs 
DV,  DR,  each  equal  to  the  obliquity  of  the  ecliptic,  which  may 
be  taken  23°  28';  join  RY,  and  on  it  describe  the  semicircle 
RTY.  With  the  sector,  opened  to  the  radius  OY  or  OR,  make 
the  arc  VU,  equal  to  the  sun’s  longitude.  When  the  longitude 
exceeds  VI  signs,  take  YI  signs  from  it,  and  set  off  the  remainder 
from  R,  round  towards  V.  Draw  UW  perpendicular  to  RV,  and 
through  W,  draw  CWL,  and  it  will  be  the  projection  of  the  circle 
of  latitude,  which  passes  through  the  moon  at  the  time  of  new 
moon.  Take  the  moon’s  latitude  from  the  scale,  and  lay  it  on  CL, 
from  C  to  M,  above  AB,  when  it  is  north ,  but  on  LC  produced, 
below  AB,  when  it  is  south.  Then  M  will  be  the  moon’s  place  at 
the  true  time  of  ecliptic  conjunction.  From  M,  draw  Mb  perpen¬ 
dicular  to  CL,  to  the  left  hand,  and  make  it  equal  to  the  moon’s 
hourly  motion  from  the  sun  in  longitude.  Draw  be  perpendicular 
to  M b,  above ,  when  the  moon’s  latitude  is  tending  north,  but  below , 
when  it  is  tending  south,  and  make  it  equal  to  the  moon’s  hourly 
motion  in  latitude.  Through  M  and  c,  draw  the  moon’s  relative 
orbit  PQ.  Make  the  proportion,  as  60  minutes  :  minutes  and 

*  For  the  manner  of  using  the  sector,  see  the  note  at  the  bottoms  of 
pages  135  and  136.  To  what  is  there  said  respecting  the  manner  of  using 
it,  may  be  added,  that  when  an  arc  greater  than  60°,  is  to  be  laid  off,  it  may 
be  done  by  applying  the  radius  of  the  circle  as  a  chord  to  the  arc,  as  many 
times  successively  as  60°  is  contained  in  the  arc  to  be  laid  off,  and  then  with 
the  sector,  laying  off  from  the  last  point,  an  arc  equal  to  the  remainder. 
When  a  very  small  arc  is  to  be  laid  off  with  a  sector,  it  is  better  to  add  some 
constant  arc  to  it,  for  instance  10°.  Then  taking  the  chord  of  the  sum  from 
the  sector,  lay  it  on  the  arc,  from  the  given  point  to  a  second  one,  and 
taking  the  chord  of  the  arc  which  was  added,  set  it  from  the  second  point 
backwards,  towards  the  first.  The  arc,  intercepted  between  the  last  point 
and  the  given  one,  will  be  the  arc  which  was  to  be  laid  off. 

43 


330 


ASTRONOMY. 


seconds  of  the  true  time  of  new  moon  :  :  moon’s  hourly  motion 
from  the  sun  on  the  relative  orbit  :  a  fourth  term.  Take  this 
fourth  term  from  the  scale,  and  lay  it  on  the  relative  orbit,  from 
M  to  the  right  hand,  and  it  will  give  the  moon’s  place  at  the  whole 
hour  next  preceding  the  time  of  new  moon.  Take  the  moon’s 
hourly  motion  from  the  sun  on  the  relative  orbit,  from  the  scale, 
and  with  it,  lay  off  equal  spaces  on  each  side  of  the  moon’s  place, 
just  found,  and  thus  obtain  the  moon’s  places  for  four  or  five  other 
hours,  contiguous  to  the  time  of  new  moon,  some  of  them  preced¬ 
ing  and  some  following  it.  When  the  time  of  new  moon  is  several 
hours  before  noon,  there  should  be  more  places  found  for  hours 
preceding  the  time  of  new  moon,  than  for  the  hours  following  it, 
and  the  contrary ,  when  the  time  of  new  moon  is  several  hours 
past  noon.  To  each  of  the  moon  s  places,  thus  found,  put  the 
number  of  the  hour. 

With  a  sector,  opened  to  the  radius  AC  or  CB,  set  off  arcs  equal 
to  the  reduced  latitude  of  the  place,  from  A  to  E  and  B  to  F, 
on  the  semicircle  above  AB  when  the  latitude  is  north ,  but  below , 
when  it  is  south,  and  join  EF.  With  the  sector,  opened  to  the 
sarhe  radius,  make  the  arcs  EG,  El,  FH  and  FK,  each  equal  to 
the  sun^s  declination,  and  join  GH  and  IK.  Bisect  vw  in  N,  and 
through  N,  draw  6  N  18,  parallel  to  EF.  Make  N  6  and  N  18, 
each  equal  to  Er  or  rF,  and  on  6  N  18,  describe  the  semicircle 
6  Y  18.  With  the  centre  N  and  radius  Nv  or  Nw,  describe  the 
circle  avxw  Take  the  intervals  between  noon  and  each  of  the 
hours  marked  on  the  relative  orbit,  and  convert  them  into  de¬ 
grees,  allowing  15°  to  each  hour,  and  they  will  be  the  hour  angles 
from  noon.  With  the  sector,  opened  to  the  radius  N  6  or  N  18, 
lay  off  from  Y,  on  the  semicircle  6  Y  18,  the  arc  being  produced 
above  6  N  18,  when  necessary,  arcs  equal  to  each  of  the  hour  an¬ 
gles,  laying  them  to  the  right,  when  the  hours  are  in  the  forenoon, 
but  to  the  left ,  when  they  are  in  the  afternoon ,  and  at  the  extremi¬ 
ty  of  each  arc,  place  the  number  of  degrees  which  it  contains. 
From  these  points,  draw  lines  parallel  to  the  universal  meridian 
DY.  Also,  from  the  same  points,  draw  lines  to  the  centre  N,  in¬ 
tersecting  the  circle  uvxw;  and  when  the  sun’s  declination  is  south , 
produce  them  to  meet  the  same  circle  on  the  other  side  of  N.  From 


AST110N0MY. 


331 


the  points  in  which  these  lines  intersect  the  circle  uvxw ,  when  the 
sun’s  declination  is  north ,  but  from  the  points  in  which,  being 
produced,  they  meet  it,  when  the  declination  is  south,  draw  lines 
parallel  to  EF,  to  meet  respectively,  the  corresponding  lines, 
drawn  parallel  to  the  universal  meridian;  and  the  points  in  which 
they  meet  will  be  the  sun’s  places,  on  the  circle  of  projection,  at 
the  hours  to  which  the  lines  correspond.  At  each  of  these  points 
place  the  number  of  the  hour  to  which  it  belongs.  The  points  6 
and  18,  are  always  the  sun’s  places  at  those  hours.  When  the 
declination  is  north,  the  point  v  is  the  sun’s  place  at  noon,  desig¬ 
nated  by  0;  but  when  the  declination  is  south,  the  point  w  is  the 
sun’s  place  at  noon.  From  the  places  of  the  moon  at  the  hours, 
marked  on  the  relative  orbit,  draw  lines  parallel  to  AB  or  EF,  to 
meet  the  lines,  produced,  if  necessary,  which  are  parallel  to  the 
universal  meridian,  and  pass  through  the  sun’s  places  at  the  same 
hours,  in  the  points  S. 

Draw  a  right  line  AC,  Fig .  58,  and  in  it  take  a  point  s.  Take 
the  distance  from  each  of  the  points  S,  in  Fig.  57,  to  the  cor¬ 
responding  place  of  the  moon  on  the  relative  orbit,  and  lay  it  on 
AC,  from  s,  to  the  right  or  left,  according  as  the  moon’s  place  is 
to  the  right  or  left  of  the  point  S,  and  at  the  extremity  of  each 
distance,  put  the  number  of  the  hour,  to  which  the  distance  cor¬ 
responds.  Through  each  of  these  points,  draw  lines  perpendicular 
to  AC.  This  may  be  most  conveniently  done,  by  drawing  through 
one  of  the  points  a  perpendicular  line,  and  then  parallel  to  this, 
drawing  lines  through  the  others.  Take  from  Fig.  57,  the  dis¬ 
tances  from  the  sun’s  place  at  each  of  the  hours,  marked  on  the 
relative  orbit,  to  the  corresponding  point  S,  and  place  them  on 
the  perpendiculars,  from  the  same  numbers  on  the  line  AC  in 
Fig,  58,  above  or  below  AC,  according  as  the  point  S  in  Fig.  57, 
is  above  or  below  the  sun’s  place.  At  the  extremities  of  these  dis¬ 
tances,  place  the  same  numbers  that  are  on  the  line  AC.  Join 
each  adjacent  two  of  these  extremities,  and  the  broken  line  thus 
formed  will  be  a  near  representation  of  the  moon’s  apparent ,  rela¬ 
tive  orbit,  and  the  points  on  it  will  be  the  moon’s  places  at  the 
hours,  denoted  by  their  numbers. 

With  the  centre  s,  and  a  radius  equal  to  the  sum  of  the  semi- 


332 


ASTRONOMY. 


diameters  of  the  sun  and  moon,  describe  arcs,  cutting  the  appa¬ 
rent  relative  orbit  in  B  and  E,  which  will  be  the  moon’s  places 
at  the  beginning  and  end  of  the  eclipse.  With  the  centres  B  and 
E  and  a  radius  greater  than  half  the  distance  of  these  points,  de¬ 
scribe  two  arcs,  cutting  each  other  in  a.  Lav  the  edge  of  a  ruler 
from  s  to  a,  and  draw  the  line  DsGn,  intersecting  the  apparent  orbit 
in  G,  which  will  be  the  moon  s  place  at  the  time  ^)f  greatest  ob¬ 
scuration.  From  the  moon’s  place  on  the  apparent  orbit  at  the 
whole  hour  next  following  the  end  of  the  eclipse,  draw  a  right 
line  LN  in  any  convenient  direction,  and  taking  any  short  dis¬ 
tance  in  the  dividers,  lay  it  over  12  times,  from  L  to  the  point  INI. 
Then  LM  is  to  be  considered  as  representing  an  hour,  divided 
into  parts  of  5  minutes  each,  which  must  be  reckoned  from  L 
towards  M.  Join  M  and  each  of  the  hour  points  on  the  apparent 
orbit.  From  the  points  B,  G  and  E,  draw  the  lines  Be,  G/i  and 
E /,  respectively  parallel  to  the  lines  joining  M  and  the  hours 
next  folloiving  those  points,  and  meeting  the  lines  joining  M  and 
the  hours  next  preceding  the  same  points,  in  the  points  c,  h  and  /. 
Draw  c&,  /ig,  and /e,  respectively  parallel  to  lines  joining  L,  and 
the  hours  next  preceding  the  points  B,  G  and  E.  Then  the 
minutes  corresponding  to  h,  connected  with  the  hour  next  pre¬ 
ceding  B,  those  corresponding  to  g,  connected  with  the  hour  next 
preceding  G,  and  those  corresponding  to  e,  connected  with  the 
hour  next  preceding  E,  will  be  the  times  of  the  beginning,  great¬ 
est  obscuration  and  end  of  the  eclipse. 

If  a  circle,  described  about  the  centre  s,  with  a  radius  equal  to 
the  difference  of  the  semidiameters  of  the  sun  and  moon,  cuts  the 
apparent  orbit,  the  eclipse  will  be  annular  or  total;  annular  when 
the  sun’s  semidiameter  is  greater  than  the  moon’s;  total  when  it 
is  less.  The  beginning  or  end  of  the  annular  or  total  eclipse,  when 
either  has  place,  may  be  found  in  the  same  manner  as  the  begin¬ 
ning  or  end  of  the  eclipse,  taking  the  points  in  which  the  circle 
cuts  the  apparent  orbit. 

About  the  centres  s  and  G,  with  radii  respectively  equal  to  the 
semidiameters  of  the  sun  and  moon,  describe  circles  to  represent 
those  bodies.  Take  the  distance  DH,  and  applying  it  to  the  scale, 
obtain  its  measure.  Then,  as  the  sun’s  semidiameter  :  measure  of 
DII  :  :  6  digits  :  digits  eclipsed. 


ASTRONOMY. 


333 


Take  the  interval  between  the  beginning  of  the  eclipse  and 
noon,  and  convert  it  into  degrees.  With  the  sector  opened  to  the 
radius  N  6,  or  N  18,  Fig.  57,  lav  off  from  Y,  on  the  semicircle 
6  Y  18,  the  arc  being  produced  if  necessary,  an  arc  containing 
this  number  of  degrees,  laying  it  to  the  right  or  left ,  according  as 
the  time  of  beginning  is  before  or  after  noon,  and  proceed  to  find 
the  sun’s  place  on  the  circle  of  projection  for  the  time  of  begin¬ 
ning,  in  the  same  manner  as  directed  above,  for  other  times. 
Mark  this  place  of  the  sun  with  the  letter  n,  and  join  C n.  Make 
the  angle  CsV,  Fig.  58,  equal  to  the  angle  BO*,  Fig .  57,  and 
join  sB.  Then  v  will  represent  the  sun’s  vertex  at  the  beginning 
of  the  eclipse,  z  the  place  at  w  hich  the  eclipse  commences,  and 
the  angle  VsB,  the  angular  distance  of  this  point  from  the  sun’s 
vertex. 

Note.  The  times  of  beginning,  &c.  obtained  by  projection,  are 
only  approximate  values.  But  when  the  construction  is  carefully 
made,  they  will  seldom  err  more  than  one  or  two  minutes. 

Exam.  1.  Required  the  times,  &c.  of  the  eclipse  of  the  sun  of 
August  27th,  1821,  at  Philadelphia. 

The  different  elements  necessary  for  the  construction  are  easily 
found,  and  are  as  follows: 

Elements  Collected. 

True  time  of  new  moon,  August,  26  d.  22  h.  15  m.  56  sec. 
Semidiameter  of  the  circle  of  projection,  55'  1"  =  55'.02 

Sun’s  longitude,  -  153°  57' 

Sun’s  declination,  north ,  -  -  -  10  4 

Moon’s  latitude,  north ,  -  -  -  341=  3. 6S 

Moon’s  hor.  mot.  from  sun,  in  long.  -  28  30  =  28.5 

Moon’s  hor.  mot.  in  lat.  tending  south ,  -  2  51  =  2.85 

Moon’s  hor.  mot.  from  sun,  on  rel.  orb.  -  28  39  =  28.65 

Fourth  term, .  7  36  =  7.6 

Sun’s  semidiameter,  -  15  52  =  15.87 

Moon’s  do. . 15  3=  15.05 

Sum  of  semidiameters  of  sun  and  moon,  -  30  55  =  30.92 

Latitude  of  Philadelphia,  reduced,  -  39  46  N. 


ASTRONOMY. 


331? 

Draw  AB,  Fig .  57,  and  take  the  point  C.  Take  55'. 02,  the 
semidiameter  of  the  circle  of  projection,  from  a  scale  of  equal 
parts,  and  with  the  centre  C,  describe  the  semicircle  ADB. 
Through  C,  and  perpendicular  to  AB,  draw  the  universal  meri¬ 
dian  TC  Y,  cutting  ADB  in  D  With  a  sector  opened  to  the  radius 
AC  or  CB,  make  the  arcs  DR,  DV,  each  equal  to  23°  28',  the 
obliquity  of  the  ecliptic;  join  RY,  and  on  it  describe  the  semicircle 
RTV.  With  the  sector  opened  to  the  radius  OR  or  OY,  make 
the  arc  VTU  equal  to  153°  57',  the  sun’s  longitude.  Draw  UW 
perpendicular  to  RY,  and  through  W,  draw  CWL.  Take  3'. 68, 
the  moon’s  latitude,  from  the  scale,  and  lay  it  on  CL,  from  C  to 
M,  above  AB,  because  the  latitude  is  north.  Draw  Mb  perpen¬ 
dicular  to  CL,  and  make  it  equal  to  28'. 5,  the  moon’s  hourly  mo¬ 
tion  from  the  sun  in  longitude.  Draw  be  perpendicular  to  M5, 
below  M 6,  because  the  latitude  is  tending  south,  and  make  it  equal 
to  2'. 85,  the  moon’s  hourly  motion  in  latitude.  Through  M  and  c, 
draw  the  moon’s  relative  orbit  PQ.  Take  7'. 6,  the  fourth  term, 
from  the  scale,  and  lay  it  on  the  relative  orbit,  from  M  to  XXII, 
the  moon’s  place  at  that  hour.  Take  28'. 65,  the  moon’s  hourly 
motion  from  the  sun  on  the  relative  orbit,  from  the  scale,  and  set  it 
over  from  XXII,  backwards  to  XXI,  XX,  and  XIX,  and  forwards 
to  XXIII,  for  the  moon  s  places  at  those  hours.  With  the  sector 
opened  to  the  radius  AC  or  CB,  make  the  arcs  AE  and  BF,  each 
equal  to  39°  46',  the  reduced  latitude  of  Philadelphia.  With  the 
sector  opened  to  the  same  radius,  make  the  arcs  EG,  El,  FII  and 
FK,  each  equal  to  10°  4',  the  sun’s  declination,  and  join  GH  and 
IK,  intersecting  the  universal  meridian  in  io  and  v.  Bisect  vw  in 
N;  through  N,  draw  6  N  18,  parallel  to  EF,  and  make  N  6,  and 
N  18,  each  equal  to  rE  or  rF.  With  the  centre  N  and  radius 
N  6  or  N  18,  describe  the  semicircle  6  Y  18,  and  with  the  same 
centre,  and  radius  Nr  or  Nw,  describe  the  circle  uvxxe.  The  in¬ 
tervals  between  noon  and  the  hours,  marked  on  the  relative  orbit, 
are  1,  2,  3,  4  and  5  hours,  and  these  in  degrees  are  15°,  30°,  45°, 
60°  and  75°.  With  the  sector  opened  to  the  radius  N  6  or  N  18, 
lay  off  these  arcs  on  the  semicircle  6  Y  18,  all  of  them  from  Y  to 
the  right  hand,  because  the  hours  are  all  in  the  forenoon.  From 
the  points  15,  30,  45,  60  and  75,  which  are  the  extremities  of  the 


ASTRONOMY. 


335 


arcs,  draw  the  lines  15,  23;  30,  22;  45,  21 ;  60, 20;  and  75, 19, 
parallel  to  the  universal  meridian  TY;  and  from  the  same  points, 
draw  lines  to  the  centre  N,  not  producing  them,  because  the  sun’s 
declination  is  north.  From  the  points  in  which  the  lines  N  15, 
N  30,  N  45,  N  60,  and  N  75,  intersect  the  circle  uvxw ,  draw  lines 
parallel  to  EF,  respectively  meeting  the  lines  15, 23;  30, 22; 
45,21;  60,20;  and  75,  19,  in  the  points  23,  22,21,20,  and  19, 
which  are  the  sun’s  places  at  those  hours.  From  the  points  XIX, 
XX,  XXI,  XXII  and  XXIII,  draw,  parallel  to  AB  or  EF,  the  lines 
XIXS,  XXS,  XXIS,  XXIIS,  and  XXIIIS,  meeting  the  lines  75, 
19;  60,20;  45,21;  30,  22;  and  15,23,  in  the  points  S.  Draw 
AC,  Fig.  58,  and  in  it  take  the  point  s.  Take  the  distances 
SXIX,  SXX,  SXXI,  SXXII,  and  SXXIII,  Fig.  57,  and  set  them 
on  the  line  AC,  Fig.  58,  from  s  to  19, 20, 21, 22  and  23,  placing 
the  first  three  to  the  right  ofs,  because  the  moon’s  places  at  those 
hours  are  to  the  right  of  the  corresponding  points  S,  and  the  other 
two  to  the  left,  because  the  moon’s  places  are  at  those  hours  to 
the  left  of  the  corresponding  points  S.  Draw  21,  XXI,  perpen¬ 
dicular  to  AC,  and  parallel  to  it,  draw  1 9,  XIX;  20,  XX;  22,  XXII; 
and  23,  XXIII.  Take  the  distances  S19,  S20,  S21,  S 22,  and 
S23,  Fig.  57,  and  set  them  in  Fig.  58,  from  19  to  XIX,  20  to 
XX,  21  to  XXI,  22  to  XXII,  and  23  to  XXIII,  setting  the  first 
two  above  AC,  because  the  points  S  are  above  the  sun’s  places, 
and  the  others  below,  because  the  points  S  are  below  the  sun’s 
places.  Join  XIX,  XX;  XX,  XXI;  XXI,  XXII;  and  XXII,  XXIII, 
for  the  apparent  relative  orbit  of  the  moon.  Take  30'. 92,  the  sum 
of  the  semidiameters  of  the  sun  and  moon,  from  the  scale,  and 
with  the  centre  s ,  describe  arcs  cutting  the  apparent  orbit  in  B 
and  E,  the  moon’s  places  at  the  beginning  and  end.  With  the 
centres  B  and  E,  and  a  radius  greater  than  half  the  distance  be¬ 
tween  them,  describe  arcs  cutting  each  other  in  a;  and  with  the 
edge  of  a  ruler,  applied  to  s  and  a,  draw  the  line  DsGH,  inter¬ 
secting  the  apparent  orbit  in  G,  the  moon’s  place  at  the  greatest 
obscuration.  From  the  point  XXIII,  in  the  apparent  orbit,  draw 
LN,  and  taking  some  short  distance  in  the  dividers,  lay  it  over 
12  times,  from  L  to  M,  and  number  the  divisions  as  in  the  figure. 
Join  M,  XIX;  M,  XX;  M,  XXI;  and  M,  XXII,  and  draw  Be  pa- 


336 


ASTRONOMY, 


rallel  to  M,  XX;  G h  parallel  to  M,  XXI;  and  E/ parallel  to  ML. 
Draw  cb  parallel  to  L,  XIX;  hg  parallel  to  L,  XX;  and  fe  parallel 
to  L,  XXII.  Then,  attending  to  the  rule,  it  is  easy  to  perceive 
that  the  beginning  of  the  eclipse  is  at  19  h.  31  m.;  the  greatest 
obscuration  at  20  h.  48  m  ;  and  the  end  at  22  h.  14  m.  Take 
15  .87,  the  sun’s  semidiameter,  from  the  scale,  and  with  the  cen¬ 
tre  5,  describe  a  circle  to  represent  the  sun,  and  with  15  .05,  the 
moon’s  semidiameter,  taken  from  the  scale,  and  the  centre  G, 
describe  another  circle,  to  represent  the  moon.  The  distance  DH, 
applied  to  the  scale,  will  be  found  to  measure  22'. 9.  Then, 
15'. 87  :  22'. 9  :  :  6  digits  :  8f  digits,  the  quantity  of  the  eclipse. 
The  interval  between  the  time  of  beginning  and  noon  is  4h.  29  m. 
which  in  degrees  is  67°  15'.  With  the  sector  opened  to  the  radius 
N  6  or  N  18,  Fig .  57,  lay  off  this  arc  on  the  semicircle  6  Y  18, 
from  Y  to  the  right  hand,  because  the  time  is  in  the  forenoon,  and 
find  n,  the  sun’s  place  at  that  time,  in  the  same  manner  as  for 
other  times.  Join  On,  and  make  the  angle  CsV,  Fig.  58,  equal 
to  BCn,  and  join  sB  The  measure  of  the  angle  BsV  is  26°, 
which  is  the  angular  distance  of  the  point  at  which  the  eclipse 
commences  from  the  sun’s  vertex  to  the  right  hand.  In  Fig.  59, 
is  a  reduced  representation  of  the  sun’s  and  moon’s  discs,  with  the 
line  sY  placed  in  a  vertical  position. 

2.  Required  to  calculate  the  elements,  and  project  an  eclipse 
of  the  sun,  for  the  latitude  and  meridian  of  Philadelphia,  that  will 
occur  in  February,  1831. 

Elements. 

True  time  of  new  moon,  February,  lid.  23  h.  57  m.  40  sec. 
Semidiam.  of  circle  of  projection,  -  57' 24"  =  57'. 4 

Sun’s  longitude,  -  323°  18 

Sun’s  declination,  south,  -  -  13  46 

Moon’s  latitude,  north,  -  -  -  42  10  =  42.17 

Moon’s  hor.  mot.  from  sun,  in  long.  -  31  4  =  31.07 

Moon’s  hor.  mot.  in  lat.  tending  south ,  -  3  4=  3.07 

Moon’s  hor.  mot.  from  sun,  on  rel.  orb.  -  31  13  =  31.22 

Fourth  term, .  30  0  =  30.00 


ASTRONOMY. 


337 

Sun’s  semidiameter,  -  l(y  14"  *=  16'.23 

Moon’s  do.  -  -  -  -  15  42  =  15.7 

Sum  of  semicfiameters,  -  -  -  -  31  56  =  31. 93 

Latitude  of  Philadelphia,  reduced,  -  39°  46 

Result  of  Projection. 

d.  h.  m. 

Beginning,  -  -  12  11  7  A.  M. 

Greatest  obscuration,  -  0  42  P.  M. 

End,  •«  -  -  2  11 

Digits  eclipsed  Ilf,  on  sun’s  south  limb. 

Eclipse  commences  about  101°,  from  the  sun’s  vertex  to  the 
right  hand. 

PROBLEM  XXXVI. 

To  Calculate  an  Eclipse  of  the  Sun ,  for  a  given  Place ,  having 
given  the  Approximate  Times ,  obtained  by  Projection. 

From  the  sun’s  longitude  and  hourly  motion,  previously  found 
for  the  approximate  time  of  new  moon,  find  his  longitude  at  the 
approximate  times  of  beginning,  greatest  obscuration,  and  end  of 
the  eclipse.  Also,  find  the  sun’s  semidiameter  and  the  apparent 
obliquity  of  the  ecliptic  for  the  approximate  time  of  new  moon. 
These  change  so  slowly  that  they  may  be  considered  the  same, 
during  the  continuance  of  the  eclipse.  Calculate  the  moon’s  lon¬ 
gitude,  latitude,  equatorial  parallax  and  semidiameter  for  the  ap¬ 
proximate  times  of  beginning,  greatest  obscuration  and  end.* 

Calculate  by  problems  XXVIII.  and  XXIX,  the  moon’s  appa¬ 
rent  longitude,  latitude,  and  augmented  semidiameter,  for  the  ap¬ 
proximate  times  of  beginning,  greatest  obscuration  and  end,  using 


*  When  great  accuracy  is  not  required,  it  will  be  sufficient  to  calculate 
the  moon’s  longitude,  latitude,  equatorial  parallax,  semidiameter,  and  hourly 
motions  in  longitude  and  latitude,  for  the  approximate  time  of  greatest  ob¬ 
scuration,  and  by  means  of  the  hourly  motions,  find  the  longitude  and  lati¬ 
tude  for  the  approximate  times  of  beginning  and  end.  The  parallax  and 
semidiameter  may,  without  material  error,  be  considered  the  same,  during 
the  eclipse 


44 


338 


ASTRONOMY. 


the  reduced  latitude  of  the  place,  and  the  difference  between  the 
reduced  parallax  of  the  moon  and  the  sun’s  parallax.  It  is  neces¬ 
sary  to  know  for  each  of  the  apparent  latitudes,  whether  it  is  in¬ 
creasing  or  decreasing.  This  may  be  determined  by  observing 
that,  when  at  the  beginning  and  end  of  any  short  interval  of  time, 
they  are  both  of  the  same  name,  the  apparent  latitude  is  increasing 
or  decreasing ,  according  as  it  is  greater  or  less  at  the  end  of  the 
interval,  than  at  the  beginning.  When  they  are  of  different  names, 
it  is  decreasing  at  the  beginning  of  the  interval,  and  increasing  at 
the  end. 

For  the  Beginning . 

Subtract  the  moon’s  apparent  longitude  at  the  approximate  time 
of  beginning  from  the  sun’s  longitude  at  the  same  time,  increasing 
the  latter  by  360°,  when  necessary,  and  call  the  remainder  G. 
Call  the  moon’s  apparent  latitude  at  the  approximate  time  of  be¬ 
ginning,  H,  the  sum  of  the  moon’s  augmented  semidiameter,  at 
the  same  time,  and  the  sun’s  semidiameter,  S,  and  the  interval 
between  the  approximate  times  of  beginning  and  greatest  obscura¬ 
tion,  T. 

Subtract  the  moon’s  apparent  longitude  at  the  approximate  time 
of  beginning,  from  its  apparent  longitude  at  the  approximate  time 
of  greatest  obscuration,  increasing  the  latter  by  360°,  when  neces¬ 
sary ;  and  do  the  same  with  the  sun’s  longitudes  at  the  same  times. 
Take  the  difference  between  the  remainders,  and  call  it  M.  When 
the  moon’s  apparent  latitudes  at  the  approximate  times  of  begin¬ 
ning  and  greatest  obscuration,  are  of  the  same  name,  take  their 
difference;  but  when  they  are  of  different  names,  take  their  sum; 
and  call  the  difference  or  sum,  N.  The  value  ofN  must  be  mark¬ 
ed  negative ,  when  the  apparent  latitude  at  the  approximate  time 
of  beginning  is  increasing ,  but  affirmative ,  when  it  is  decreasing. 

Add  together,  the  logarithm  of  H,  the  logarithm  of  N,  and  the 
arithmetical  complement  of  the  logarithm  of  M,  and  the  result, 
rejecting  the  tens  in  the  index,  will  be  the  logarithm  of  a  small 
arc  V.  Apply  V,  according  to  its  sign,  to  G,  and  call  the  result 
W.  To  the  logarithm  of  the  sum  of  S  and  H,  add  the  logarithm 


ASTRONOMY.  339 

of  their  difference,  and  divide  the  sum  by  2;  the  result  will  be  the 
logarithm  of  an  arc  L. 

Add  together,  the  logarithm  of  T,  the  logarithm  of  the  sum  of 
G  and  L,  the  logarithm  of  their  difference,  the  arithmetical  com¬ 
plement  of  the  logarithm  of  2  M,  and  the  arithmetical  complement 
of  the  logarithm  of  W,  and  the  result,  rejecting  the  tens  in  the 
index,  will  be  the  logarithm  of  a  correction ,  which,  added  to  the 
approximate  time  of  beginning,  when  G  is  greater  than  L,  but 
subtracted ,  when  G  is  less  than  L,  will  give  the  true  time  of  be¬ 
ginning  very  nearly. 

For  the  End. 

Subtract  the  sun’s  longitude  at  the  approximate  time  of  the  end, 
from  the  moon’s  apparent  longitude  at  the  same  time,  increasing 
the  latter  by  360°.  when  necessary,  and  call  the  remainder  G. 
Call  the  moon’s  apparent  latitude  at  the  approximate  time  of  the 
end,  H,  the  sum  of  the  moon’s  augmented  semidiameter,  at  the 
same  time,  and  the  sun’s  semidiameter,  S,  and  the  interval  be¬ 
tween  the  approximate  times  of  greatest  obscuration  and  end,  T. 

Subtract  the  moon’s  apparent  longitude  at  the  approximate 
time  of  greatest  obscuration,  from  its  apparent  longitude  at  the 
approximate  time  of  the  end,  increasing  the  latter  by  360°,  when 
necessary;  and  do  the  same  with  the  sup’s  longitudes  at  the  same 
times.  Take  the  difference  between  the  remainders,  and  call  it  M. 
When  the  moon’s  apparent  latitudes  at  the  approximate  times  of 
greatest  obscuration  and  end  are  of  the  same  name,  take  their 
difference;  but  when  they  are  of  different  names,  take  their  sum; 
and  call  the  difference  or  sum,  N.  The  value  of  N  must  be  mark¬ 
ed  affirmative ,  when  the  apparent  latitude  at  the  approximate  time 
of  the  end  is  increasing ,  but  negative ,  when  it  is  decreasing . 

Find  the  quantities  V,  W,  L,  and  the  correction ,  as  directed 
for  the  beginning.  The  correction,  added  to  the  approximate  time 
of  the  end,  when  G  is  less  than  L,  but  subtractedi  when  G  is 
greater  than  L,  will  give  the  true  time  of  the  end. 


340 


ASTRONOMY. 


For  the  Greatest  Obscuration ,  and  Quantity  of  the  Eclipse . 

Subtract  the  moon’s  apparent  longitude  at  the  approximate 
time  of  greatest  obscuration,  from  the  sun’s  longitude  at  the  same 
time,  increasing  the  latter  by  360°,  when  necessary;  and  when  the 
remainder  is  a  small  arc,  mark  it  affirmative ,  and  call  it  G;  but 
when  it  is  near  to  360°,  subtract  it  from  360°,  and  marking  the 
second  remainder  negative ,  call  it  G.  Call  the  moon’s  apparent 
latitude  at  the  approximate  ti me  of  greatest  obscuration  H,  the 
sum  of  the  moon’s  augmented  semidiameter,  at  the  same  time, 
and  the  sun’s  semidiameter,  S,  and  the  interval  between  the  ap¬ 
proximate  times  of  beginning  and  end,  T. 

Subtract  the  moon’s  apparent  longitude  at  the  approximate 
time  of  beginning,  from  its  apparent  longitude  at  the  approximate 
time  of  the  end,  increasing  the  latter  by  360°,  when  necessary; 
arid  do  the  same  with  the  sun’s  longitudes  at  the  same  times. 
Take  the  difference  between  the  remainders,  and  call  it  M.  When 
the  moon’s  apparent  latitudes,  at  the  approximate  times  of  begin  ¬ 
ning  and  end,  are  of  the  same  name,  take  their  difference;  but 
when  they  are  of  different  names,  take  their  sum;  and  call  the  dif¬ 
ference  or  sum,  N.  The  value  of  N  must  be  marked  negative , 
when  the  apparent  latitude  at  the  approximate  time  of  greatest 
obscuration  is  increasing ,  but  affirmative ,  when  it  is  decreasing. 

To  the  logarithm  of  N,  add  the  arithmetical  complement  of  the 
logarithm  of  M,  and  the  result  will  be  the  Tangent  of  an  arc  I, 
which  must  be  taken  out  according  to  the  sign,  but  less  than  180°. 
To  the  Tangent  of  I,  add  the  logarithm  of  H,  and  the  result,  re¬ 
jecting  the  tens  in  the  index,  will  be  the  logarithm  of  a  small  arc 
V.  Take  the  sum  of  G  and  V,  attending  to  their  signs,  and  call 
it  W.  Add  together,  the  logarithm  of  W  and  the  Cosine  of  I, 
taken  affirmative,  and  call  the  resulting  logarithm  X.  Add  to¬ 
gether,  the  logarithm  X,  the  Cosine  of  I,  taken  affirmative,  the 
logarithm  of  T,  and  the  arithmetical  complement  of  the  logarithm 
f  M,  and  the  result,  rejecting  the  tens  in  the  index,  will  be  the 
logarithm  of  a  correction ,  which  applied,  according  to  its  sign,  to 
the  approximate  time  of  greatest  obscuration,  will  give  the  true 
time. 


ASTRONOMY. 


341 


Add  together,  the  logarithm  X,  and  the  Tangent  of  I,  and  the 
result,  rejecting  the  tens  in  the  index,  will  be  the  logarithm  of  a 
small  arc  Y.  Apply  Y  to  S,  according  to  its  sign,  and  call  the 
result  S'.  To  the  logarithm  of  H,  add  the  Cosine  of  I,  taken  af¬ 
firmative,  and  the  result,  rejecting  the  tens  in  the  index,  will  be 
the  logarithm  of  an  arc  H'.  Add  together,  the  constant  logarithm 
0.77815,  the  logarithm  of  the  difference  between  S'  and  IT,  and 
the  arithmetical  complement  of  the  logarithm  of  the  sun’s  semi- 
diameter,  and  the  result,  rejecting  the  tens  in  the  index,  will  bfr 
the  logarithm  of  the  digits  eclipsed. 

Note  1.  If  Y  be  applied,  with  a  contrary  sign,  to  IT,  it  will 
give  the  apparent  distance  of  the  centres  of  the  sun  and  moon,  at 
the  time  of  greatest  obscuration.  When  this  distance  is  less  than 
the  difference  between  the  sun’s  semidiameter  and  the  augmented 
semidiameter  of  the  moon,  the  eclipse  is  either  annular  or  total; 
annular ,  when  the  sun’s  semidiameter  is  the  greater  of  the  two; 
total,  when  it  is  the  less. 

2.  When  the  point  of  the  sun’s  disc,  at  which  the  eclipse  com¬ 
mences,  is  required  with  greater  accuracy  than  is  given  by  the 
projection,  it  may  be  obtained  by  the  formulae  in  chapter  XI, 
art.  76. 

3.  Supposing  the  longitude,  &c.  to  be  accurate,  the  times  ob¬ 
tained  by  this  problem  will  be  true,  within  a  few  seconds.  When 
greater  accuracy  is  required,  the  calculation  may  be  made  by  the 
formulae  in  chap.  XI,  articles  67  to  76. 

Exam.  1.  The  approximate  time  of  the  beginning  of  the  eclipse 
of  the  sun,  on  the  27th  of  August,  1821,  found  by  projection  for 
the  latitude  and  meridian  of  Philadelphia,  is  7  h.  31m.  A.  M.; 
greatest  obscuration,  8  h.  48  m.  A.  M.;  and  end,  10  h.  14  m.  A. 
M.  .Required  the  true  times  and  quantity  of  the  eclipse. 

By  reducing  each  of  the  given  times  to  mean  time  at  Green¬ 
wich,  and  calculating  the  sun’s  and  moon’s  longitudes,  &c.  for  those 
times;  and  then  calculating  the  parallaxes,  the  following  quanti¬ 
fies  will  be  obtained: 


342 


ASTRONOMY, 


Approx,  time  of  Approx,  time  of  Approx,  time 
Beginning.  Greatest  Obscur.  of  End. 

7  b.  31m.  8h.  48  m.  10  h.  14  m. 

Sun’s  true  longitude,  153°  50'  36"  153°  53'  42"  153°  57'  10'' 

Sun’s  semidiameter,  15  52  15  52  15  52 

Moon’s  appar.  long.  153  19  52  153  51  53  154  24  21 

Moon’s  appar.  lat.  3  59  S.  8  27  S.  15  40S 

Moon’s  augm.  semid.  1510  1513  1514 

For  the  Beginning. 

G  =  30'  44"  =  1844";  H  =  3'  59"  =  239";  S  =  31'2"  = 
1862";  T  =  1  h.  17  m.  =  4620  sec.;  M  =  28'  55"  =  1735"; 
and  N  =  —  4'  28"  =  —  268". 


H 

239” 

log.  2.37840 

N  - 

—  268 

-  log.  — 2.42813 

M 

1735 

Ar.  Co.  log.  6.76070 

V 

—  37 

log.  — 1.56723 

G 

1844 

W 

-  1807" 

S  +  H  - 

.  2101" 

log.  3.32243 

S  — H 

1623 

log.  3.21032 

.  ..  . ..  > 

2)6.532 75 

L 

1847" 

log.  3.26637 

T 

4620  sec.  log.  3.66464 

G  +  L  - 

3691" 

log.  3.56714 

L  — G 

3 

log.  0.47712 

2  M 

3470 

Ar.  Co.  log.  6.45967 

W 

1807 

Ar.  Co.  log.  6.74304 

Correction, 

8  sec.  log.  0.91161 

Approx,  time  of  Begi 

n.  7h.  31  m.  0 

True  time  of  Begin.  7h.  30m.  52  sec. 


ASTRONOMY. 

343 

For  the  End. 

G=  1631”;  H  = 

:  940"; S  =  1866 

T  =  5160  sec.;  M  = 

1740”;  and  N  =  + 

433". 

H 

940" 

log.  2.97313 

N  - 

433 

log.  2.63649 

M 

.  -  1740 

Ar.  Co.  log.  6.75945 

V 

+  234 

log.  2.36907 

G 

-  1631 

W 

1865 

S  +  H 

2806" 

log.  3.44809 

S  —  H 

926 

log.  2.96661 

2)6.41470 

L 

1612" 

log.  3.20735 

T 

5160  sec.  log.  3.71265 

G  +  L 

3243" 

log.  3.51095 

G  — L 

19 

log.  1.27875 

2  M 

3480 

Ar.  Co.  log.  6.45842 

W 

1865 

Ar.  Co.  log.  6.72932 

Correction, 

49  sec.  log.  1.69009 

Approx,  time  of  End,  lOli.  14m.  Osec. 


True  time  of  End,  10b.  13m.  11  sec. 


ASTRONOMY 


344 


For  Greatest  Obscuration ,  and  Digits  Eclipsed . 

0  ==  4-  109";  H  =  507";  S  »  1865";  T  =  9780  sec. 
M  =  3475";  and  N  =  —  701". 


N 

M 

—  701'  log.— 2.84572 

3475  Ar.  Co.  log.  6.45905 

I 

H 

16S°36' 

507"  - 

tan.  —  9.30477 
log.  2.70501 

• 

■ 

i 

i 

— 102  - 

+  109 

log.— 2.00978 

W 

I 

-  4-  7"  - 

168°  36' 

log.  0.84510 
cos.  9.99135 

i 

•  . 

'  * 

X  10.83645 
cos.  9.99135 
9780  sec.  log.  3.99034 

3475"  Ar.  Co.  log.  6.45905 

Correction, 

Approx.  time,G.  Obscur. 

4-  19  sec. 
8b.  48m.  0  sec. 

log.  1.27719 

True  time,  G.  Obscur. 

8h.48m.  19  sec. 

I 

- 

X.  10.83645 
tan.  —  9.30477 

• 

i 

• 

• 

tx  CO 

—  1"  - 

1865 

log.  — 0.14122 

S' 

1864 

H 

507"  -  log.  2.70501 

-  -  Ar.  Co.  cos.  0.00865 

H' 

-  517"  - 

log.  2.71366 

S'  —  H' 

Sun’s  semidiameter, 

0.77815 

1347"  -  log.  3.12937 

952  Ar.  Co.  log.  7.02136 

Digits  eclipsed, 

8.5 

log.  0.92888 

ASTRONOMY. 


345 


2.  The  approximate  time  of  the  beginning  of  the  eclipse  of  the 
sun,  that  will  occur  on  the  12th  of  February,  1831,  found  by 
projection,  for  the  latitude  and  meridian  of  Philadelphia,  is  1 1  h. 
7m.  A.  M  ;  greatest  obscuration,  Oh.  42  m.  P.  M.;  and  end, 
2  h.  11  m.  P.  M.  Required  the  true  times,  and  the  quantity  of 
the  eclipse.  Jins,  Beginning,  11  h.  7  m.  12  sec.  A.  M.;  greatest 
obscuration,  Oh.  41  m.  29 sec.  P.  M.;  end,  2  h.  10m.  32  sec.  P. 
M.;  digits  eclipsed,  lli. 

PROBLEM  XXXVII. 

To  find  by  Projection ,  the  Latitudes  and  Longitudes  of  the 
Places  at  which  an  Eclipse  of  the  Sun  is  Central ,  for  different  times 
during  the  continuance  of  the  Central  Eclipse . 

Draw  AB,  Fig.  60,  and  perpendicular  to  it,  draw  the  univer¬ 
sal  meridian  DCY.  With  the  centre  C,  and  a  radius  equal  to  the 
semidiameter  of  the  circle  of  projection,  describe  the  circle  of 
projection  ADBY;  and  proceed  as  directed  in  prob.  XXXV,  to 
draw  the  moon’s  relative  orbit,  and  to  find  the  moon’s  places  on 
it  at  such  whole  hours  as  will  fall  on  the  circle  of  projection,  or 
near  to  it.  Or  when  the  eclipse  has  been  previously  projected  for 
a  particular  place,  this  part  may  be  obtained  by  pricking  it  off 
from  that  projection.  Divide  the  hour  spaces  on  the  relative  orbit 
into  five  minute  or  minute  spaces.  With  a  sector  opened  to  the 
distance  AC  or  CB,  make  the  arc  DP  equal  to  the  sun’s  declina¬ 
tion,  laying  it  to  the  left,  when  the  declination  is  north,  but  to  the 
right,  when  it  is  south;  draw  PC p,  and  EC  perpendicular  to  it. 

For*  the  Place  at  which  the  Sun  is  Centrally  Eclipsed,  on  the  Me¬ 
ridian. 

From  the  point  n,  in  which  the  relative  orbit  intersects  the 
universal  meridian,  draw  nq  parallel  to  AB.  Then  E^,  measured 
with  the  sector,*  will  give  the  latitude  of  the  place,  which  will 
be  north  or  south,  according  as  q  is  above  or  below  E.  The  inter¬ 
val  between  noon  and  the  time  on  the  relative  orbit,  correspond- 


*  See  note,  pages  135  and  13(5. 


34G 


ASTRONOMY. 


ing  to  n,  converted  into  degrees,  will  give  the  longitude  of  the 
place,  reckoned  from  the  meridian  of  the  place  for  which  the  pro¬ 
jection  is  made;  and  it  will  be  east  or  west,  according  as  the  time, 
reckoned  astronomically,  is  more  or  Zessthan  12  hours. 

For  the  Places  at  which  the  Eclipse  Commences  or  Ceases  to  be 

Central . 

The  central  eclipse  commences  when  the  moon’s  centre  is  at  a, 
and  ends  when  it  is  at  e.  From  the  points  a  and  e,  draw  the  lines 
af  and  et,  parallel  to  AB,  meeting  the  universal  meridian  in /and 
t;  and  from  or  through/  and  Z,  draw  frs  and  utv,  parallel  to  EC, 
cutting  or  meeting  the  line  P p  in  r  and  u  Then  Es,  measured 
with  the  sector,  will  give  the  latitude  of  the  place  at  which  the 
eclipse  is  central,  when  the  moon’s  centre  is  at  a,  and  Er  will 
give  the  latitude  of  the  place  at  which  it  is  central  when  the  moon’s 
centre  is  at  e .  These  latitudes  will  be  north  or  south,  according  as 
the  points  s  and  v  are  above  or  below  E. 

From  a  and  e,  draw  ad  and  ek,  parallel  to  the  universal  meri¬ 
dian.,  Take  the  distance  fr,  and  lay  it  from  d  to  g,  on  ad,  pro¬ 
duced  if  necessary,  above  or  below  AB,  according  as  /is  to  the 
right  or  left  of  Pp.  Take  the  distance  ut,  and  lay  it  in  like  man¬ 
ner  from  k  to  w ,  above  or  below  AB,  according  as  Z  is  to  the  right 
or  left  of  P p  Through  g  and  ic,  draw  the  lines  Cgh  and  Cwx. 

By  means  of  the  sector,  measure  the  arc  Y/i,  and  call  it  west  or 
east ,  according  as  h  is  to  the  right  or  left  of  the  universal  meridian. 
Take  the  interval  between  noon  and  the  time  on  the  relative  orbit, 
corresponding  to  a,  and  convert  it  into  degrees,  and  it  will  give 
the  hour  angle,  which  must  be  marked  west  or  east,  according  as 
the  time  is  more  or  less  than  12  hours.  When  the  arc  and  hour 
angle  are  of  the  same  name,  take  their  difference;  and  if  the  arc 
is  the  greater  of  the  two,  mark  the  difference  also  with  the  same 
name;  but  if  the  arc  is  the  less  of  the  two,  mark  the  difference  with 
a  contrary  name.  When  the  arc  and  hour  angle  are  of  different 
names,  take  their  sum ,  and  mark  it  with  the  same  name  as  the 
arc.  The  result  in  either  case  will  be  the  longitude  of  the  place 
at  which  the  eclipse  is  central  when  the  moon’s  centre  is  at  a , 


ASTRONOMY.  347 

reckoned  from  the  meridian  of  the  place  for  which  the  construc¬ 
tion  is  made. 

In  like  manner,  with  the  arc  Ya?,  and  the  time  the  moon’s  cen¬ 
tre  is  at  e,  find  the  longitude  of  the  place  which  then  has  the  cen¬ 
tral  eclipse. 

|r  .  \  • 

For  the  Place  at  which  the  Eclipse  is  central  at  any  other  time 
during  the  continuance  of  the  Central  Eclipse . 

Let  T  be  the  moon’s  place  at  the  given  time.  Through  T,  draw 
ZMN  parallel  to  the  universal  meridian,  and  TK  parallel  to  AB. 
Take  the  distance  MN,  and  witlr  the  centre  C,  describe  an  arc, 
cutting  TK  in  K,  on  the  left  of  the  universal  meridian.  Through 

K,  draw  pKz,  parallel  to  EC.  Then  Ei/,  measured  with  the  sector, 
will  give  the  latitude  of  the  place  which  has  a  central  eclipse 
when  the  moon’s  centre  is  at  T;  the  latitude  being  north  or  south, 
according  as  y  is  above  or  below  E. 

Take  the  distance  Kz ,  and  lay  it  on  NMZ,  from  M  to  Z,  below 

AB,  when  K  is  to  the  left  of  Pp,  as  is  generally  the  case;  but  above 
AB,  if  K  is  to  the  right  of  P p.  Through  Z,  draw  CZS.  Then, 
with  the  arc  YS  and  the  time  the  moon’s  centre  is  at  T,  find  the 
longitude  of  the  place  at  which  the  eclipse  is  central  in  the  same 
manner  as  directed  above,  for  the  arc  Y h  and  the  time  of  the 
moon’s  centre  being  at  a. 

Note.  From  the  latitudes  and  longitudes  thus  determined  for 
a  number  of  times  during  the  eclipse,  the  path  of  the  central 
eclipse  may  be  drawn  on  a  map.  These  latitudes  and  longitudes, 
determined  by  projection,  can  not  be  depended  on  as  accurate. 
But  when  the  construction  is  carefully  performed,  they  will  sel¬ 
dom  err  more  than  15'  or  20',  and  will  therefore  serve  to  ascer¬ 
tain  nearly  the  places  at  which  the  eclipse  will  be  central. 

Exam.  Required  the  latitudes  and  longitudes  of  the  places  at 
which  the  eclipse  of  the  sun  of  August,  1821,  will  be  central  on 
the  meridian,  will  commence  and  cease  to  be  central,  and  at 
which  it  will  be  central  at  the  whole  hours  during  its  continuance, 
reckoned  on  the  meridian  of  Philadelphia. 


348 


ASTRONOMY. 


This  is  the  eclipse,  projected  in  the  first  example  of  prob. 
XXXV.  Fig.  60,  contains  all  the  lines  necessary  for  determining 
the  latitudes  and  longitudes  required  in  this  example;  and  taken 
in  connection  with  the  above  rule,  it  is  sufficiently  plain  without 
further  explanation.  The  latitudes  and  longitudes  obtained  from 
it  are  as  follows: 


Beginning  centr.  eclipse, 

21  h.  - 

22  h.  - 
On  the  Meridian, 

23  h.  - 
Oh. 

End  centr.  eclipse, 


'  Lat. 

30°  45'  N. 
29  20  N. 
17  15  N. 

14  40  N. 
3  15  N. 

15  40  S. 
23  0  S. 


Long,  from  Philad. 
41°  45'  W. 

4  50  E. 

24  0  E. 

26  45  E. 

36  15  E. 

59  15  E. 

81  10  E. 


PROBLEM  XXXVIII. 


To  Project  an  Occultation  of  a  Fixed  Star  by  the  Moon^for  a 
given  Place. 


The  times  of  the  conjunctions  of  the  moon  with  such  stars  as 
may  suffer  occultations,  somewhere  on  the  earth,  are  given  in  the 
Nautical  Almanac,  on  the  first  page  of  each  month.  Thus,  on  the 
first  page  of  table  LV,  the  line,  3  d.  17  h.  47  m.  D  a  rr^ ,  means 
that  the  moon  is  in  conjunction  with  a  Virginis ,  on  the  3rd  day 
of  the  month,  at  17  h.  47  m.  apparent  time  at  Greenwich.  When 
the  occultation  is  visible  at  Greenwich,  the  times  of  beginning 
and  end,  or  which  is  the  same,  of  Immersion  and  Emersion ,  are 
given,  instead  of  the  time  of  conjunction. 

Find  by  prob.  XIV,  the  star’s  mean  longitude,  latitude,  right 
ascension  and  declination  for  the  day  of  the  occultation.  For  the 
time  of  conjunction,  find  by  prob.  XII,  the  moon’s  latitude,  equa¬ 
torial  parallax,  semidiameter,  and  hourly  motions  in  longitude 
and  latitude;  also,  find  for  the  same  time,  the  sun’s  longitude  and 
right  ascension.  When  the  time  of  conjunction  is  not  given,  take 
from  the  Nautical  Almanac,  the  two  longitudes  of  the  moon  next 
less  than  the  star’s  longitude,  and  the  two  next  greater.  Then, 


ASTRONOMY. 


349 


with  them  and  the  star’s  longitude,  taken  four  times,  instead  of 
the  sun’s  longitudes,  the  time  of  conjunction  may  be  found  in  the 
same  manner  as  is  directed  in  prob.  XXXI,  for  finding  the  time 
of  new  moon;  except  that  T  must  be  found  by  proportion;  thus, 
the  quantity  A  :  the  2nd  distance  : :  12  h.  :  T. 

Reduce  the  time  of  conjunction  to  the  meridian  of  the  given 
place.  When  the  latitudes  of  the  moon  and  star  are  of  the  same 
name,  take  their  difference ,  and  it  will  be  the  moon’s  distance 
from  the  star  in  latitude;  if  the  moon’s  latitude  is  the  greater  of 
the  two,  this  distance  must  be  marked  with  the  same  name  as  the 
latitudes;  but  if  the  moon’s  latitude  is  the  less  of  the  two,  the  dis¬ 
tance  must  be  marked  with  the  contrary  name.  When  the  lati¬ 
tudes  are  of  different  names,  their  sum  will  be  the  moon’s  distance 
from  the  star  in  latitude,  and  it  must  be  marked  with  the  same 
name  as  the  moon’s  latitude.  Subtract  the  sun’s  right  ascension 
from  the  right  ascension  of  the  star,  expressed  in  time,  increasing 
the  latter  by  24  hours  when  necessary,  and  the  remainder  will  be 
the  time  of  the  star’s  passage  over  the  meridian.*  Make  the  pro¬ 
portion,  as  60  m.  :  the  minutes  and  seconds  of  the  time  of  con¬ 
junction  of  the  moon  and  star  : :  the  moon’s  hourly  motion  in 
longitude  :  a  fourth  term. 

Draw  AB,  Fig.  61,  and  CT  perpendicular  to  it.  With  the 
centre  C  and  a  radius  equal  to  the  moon’s  parallax,  describe  the 
semicircle  ADB,  for  the  northern  half  of  the  circle  of  projection. 
When  the  latitude  of  the  place  is  south ,  describe  the  whole  circle. 
Make  the  arcs  DR  and  DV,  each  equal  to  23°  28',  the  obliquity 
of  the  ecliptic;  join  RV,  and  with  the  centre  0  and  radius  OR, 
describe  the  circle  RTV.  Make  the  arc  VU  equal  to  the  star’s 
longitude,  setting  it  from  V,  in  the  direction  VUT,  and  draw  Urn 
parallel  to  CT.  Make  the  arc  Dp  equal  to  the  star’s  declination, 
and  draw  pa  parallel  to  RV.  With  the  centre  C  and  radius  Ca, 
describe  the  arc  aq ,  meeting  Urn,  produced  if  necessary,  in  9,  and 
through  <7,  draw  the  circle  of  latitude  CL.  Take  the  moon’s  dis¬ 
tance  from  the  star  in  latitude,  from  the  scale,  and  when  the  dis- 

*  This  is  not  accurately  the  time  of  the  star’s  passage  over  the  meridian, 
but  it  is  nearly  so.  It  is  the  star’s  distance  from  the  sun  in  right  ascension 
at  the  time  of  conjunction,  which  is  the  quantity  wanted  in  the  projection. 


350 


ASTRONOMY. 


tance  is  north ,  set  it  on  CL,  from  C  to  M,  above  AB;  but  when 
the  distance  is  south ,  it  must  be  set  on  LC  produced,  below  AB. 
Draw  Mb  perpendicular  to  CL,  and  make  it  equal  to  the  moon’s 
hourly  motion  in  longitude.  Draw  be  parallel  to  CL,  above  or 
below  M6,  according  as  the  moon’s  hourly  motion  in  latitude  is 
tending  north  or  south ,  and  make  it  equal  to  the  hourly  motion  in 
latitude.  Through  M  and  c,  draw  the  moon’s  relative  orbit  PQ. 
Take  the  fourth  term  from  the  scale,  and  set  it  on  the  relative 
orbit,  from  M  to  the  right  hand,  for  the  moon’s  place  at  the  whole 
hour,  next  preceding  the  time  of  conjunction.  With  the  moon’s 
hourly  motion  in  longitude,  in  the  dividers,  set  off  on  the  relative 
orbit,  one  or  two  equal  spaces  on  each  side  of  this  point,  to  obtain 
the  moon’s  places  at  some  of  the  contiguous  hours,  and  mark  each 
point  with  the  proper  number  of  the  hour.  Make  the  arcs  AE 
and  BF,  each  equal  to  the  latitude  of  the  place,  and  EG,  El, 
FH  and  FK,  each  equal  to  the  star’s  declination,  and  join  GH, 
EF  and  IK.  Bisect  vw  in  N,  and  through  N,  draw  XNZ  paral¬ 
lel  to  EF.  With  the  centre  N,  and  a  radius  equal  to  rE,  describe 
the  semicircle  XYZ;  and  with  the  same  centre  and  the  radius  Nv, 
describe  the  circle  uvxw.  Take  the  intervals  between  the  time 
of  the  star’s  passage  over  the  meridian  and  each  of  the  hours  mark¬ 
ed  on  the  moon’s  orbit,  and  the  results  converted  into  degrees, 
will  give  the  hour  angles  for  the  star,  at  those  hours  respectively. 
Set  off  from  Y,  on  the  semicircle  XYZ,  produced  if  necessary, 
arcs  equal  to  these  hour  angles,  setting  them  to  the  right  for  the 
hours  on  the  orbit,  which  are  earlier  than  the  time  of  the  star’s 
passage  over  the  meridian,  but  to  the  left ,  for  the  hours  which  are 
later.  From  the  extremities  of  the  arcs,  draw  lines  parallel  to  the 
meridian  CD,  and  others  to  the  centre  N,  producing  the  latter 
when  the  star’s  declination  is  south ,  to  meet  the  circle  uvxw  on  the 
opposite  side  of  N.  From  the  points  in  which  the  lines  drawn  to 
the  centre  N,  intersect  the  circle  itimr,  when  the  declination  of 
the  star  is  north ,  or  from  the  points  in  which  they  meet  it  on  the 
opposite  side  of  N,  when  the  declination  is  south ,  draw  lines  pa¬ 
rallel  to  XZ,  to  meet  respectively  the  corresponding  lines  drawn 
from  the  extremities  of  the  arcs,  parallel  to  CD;  and  the  points  in 
w  hich  they  meet  will  be  the  star’s  places  on  the  circle  of  projec- 


ASTRONOMY. 


3d  1 


tion,  at  the  hours  to  which  the  arcs  appertain.  Mark  each  of 
these  points  with  the  proper  number  of  the  hour.  Through  each 
of  the  moon’s  places  at  the  hours  marked  on  the  relative  orbit, 
draw  lines  parallel  to  EF,  to  meet  respectively  the  lines  which 
are  parallel  to  CD,  and  pass  through  the  star’s  places  at  the  same 
hours,  in  the  points  S.  Draw  a  right  line  AC,  Fig.  62,  and  in  it 
take  a  point  s,  to  represent  a  fixed  position  of  the  star’s  place. 
Transfer  the  distances  between  the  points  S  and  the  moon  and 
star’s  places  in  Fig.  61,  to  Fig.  62,  and  draw  the  moon’s  appa¬ 
rent  relative  orbit,  in  the  same  manner  as  directed  for  the  moon 
and  sun  in  prob.  XXXV.  With  the  centre  s  and  a  radius  equal  to 
the  moon’s  semidiameter,  describe  arcs  cutting  the  apparent  orbit 
in  I  and  E,  the  moon’s  places  at  the  immersion  and  emersion; 
and  about  the  points  I  and  E,  with  the  same  radius,  describe  cir¬ 
cles  to  represent  the  moon’s  disc.  From  the  moon’s  place  on  the 
apparent  orbit,  at  the  hour  next  following  the  emersion,  draw  a 
right  line  LN  in  any  convenient  direction,  and  lay  off  from  L  to 
M,  twelve  equal  spaces,  to  represent  intervals  of  five  minutes 
each,  and  number  them  as  in  the  figure.  Join  M  and  the  moon’s 
places  at  the  hours  marked  on  the  apparent  orbit.  From  the 
points  I  and  E,  draw  lines  respectively  parallel  to  the  lines  join¬ 
ing  M  and  the  moon’s  places  at  the  hours  next  following  the 
points  I  and  E,  and  meeting  the  lines  joining  M  and  the  moon’s 
places  at  the  next  preceding  hours,  in  the  points  h  and/.  From 
h,  draw  hi  parallel  to  a  line  joining  L  and  the  moon’s  place  at  the 
hour  next  preceding  I;  and  from  /,  draw  fe  parallel  to  the  line 
joining  L  and  the  moon’s  place  at  the  hour  next  preceding  E. 
Then  the  minutes  corresponding  to  i,  connected  with  the  hour  next 
preceding  I,  will  be  the  time  of  the  immersion;  and  the  minutes 
corresponding  to  e,  connected  with  the  hour  next  preceding  E, 
will  be  the  time  of  emersion.  Take  the  intervals  between  the 
time  of  the  star’s  passage  over  the  meridian  and  the  time  of  im¬ 
mersion  and  emersion,  and  converting  them  into  degrees,  find  in 
Fig.  61,  the  points  n  and  t,  the  star’s  places  at  those  times,  in  the 
same  manner  as  directed  for  the  whole  hours.  Make  the  angles  nsC 
and  tsC,  Fig  62,  respectively  equal  to  the  angles  nCB  and  tCB, 
Fig.  61.  Through  I,  draw  bv,  parallel  to  sn ,  and  through  E,  draw 


353 


ASTRONOMY. 


b'v'  parallel  to  st;  then  bv  and  b'v'  will  represent  vertical  circles 
passing  through  the  moon's  centre  at  the  times  of  immersion  and 
emersion.  The  angles  vis  and  v'Es  will  be  the  angular  distances 
from  the  moon’s  vertex  at  which  the  immersion  and  emersion  take 
place. 

If  it  is  required  to  find  the  moon’s  phase,  and  its  position  with 
regard  to  a  vertical  circle  passing  through  the  centre  at  the  times 
of  immersion  and  emersion,  it  may  be  done  with  sufficient  accu¬ 
racy  as  follows:  Subtract  the  sun’s  longitude  from  the  longitude 
of  the  star,  increasing  the  latter  by  360°,  when  necessary,  and 
make  the  arc  Vd,  Fig.  61,  equal  to  the  remainder,  setting  it  round 
in  the  direction  VTd,  and  join  dO.  Make  the  arc  Yh  equal  to  the 
moon’s  latitude,  above  RV  when  the  latitude  is  north ,  but  below 
when  it  is  south ,  and  draw  gh  parallel  to  RV,  intersecting  dO, 
produced  if  necessary,  in  e,  and  CT  in  z.  Make  zs  equal  OR, 
and  join  se.  Make  the  angle  LC k  equal  to  zse ,  and  on  the  same 
side  of  LC  that  se  is  of  CD.  Make  the  angle  ks C,  Fig.  62,  equal 
to  &CB,  Fig.  61,  and  through  1  and  E,  draw gd  and  g'd'  parallel 
to  ks;  also,  through  the  same  points  draw  mr  and  mV  perpendicu¬ 
lar  to  gd  and  g'd'.  Then  the  points  d  and  g  will  designate  the 
positions  of  the  moon’s  cusps  with  respect  to  the  vertical  circle  bvf 
at  the  immersion,  and  the  points  d'  and  g',  the  same  with  respect 
to  b'v\  at  the  emersion.  Make  the  arcs  ru  and  rV,  each  equal  to 
the  remainder  obtained  above,  by  subtracting  the  sun’s  longitude 
from  that  of  the  star,  and  draw  uc  and  u'c'  parallel  to  dg  and  d'g\ 
and  meeting  mr  and  mV  in  the  points  c  and  c'.  Describe  the  cir¬ 
cular  arcs  deg  and  d'c'g'.  Then,  when  the  above  remainder  is 
less  than  180°,  degr  and  d'c'g'r'  will  be  near  representations  of  the 
enlightened  disc  of  the  moon;  but  when  the  remainder  is  greater 
than  180°,  dmgc  and  d'm'g'c'  will  be  representations  of  the  en¬ 
lightened  disc.  For  the  example  to  which  the  figures  are  adapted, 
Fig.  63,  represents  the  position  of  the  moon’s  phase  and  of  the 
star  at  the  emersion,  with  respect  to  the  vertical  circle  6v,  placed 
in  a  vertical  position,  and  Fig .  64,  does  the  same  for  the  emer¬ 
sion. 


ASTRONOMY. 


353 


Note  1.  No  notice  has  been  taken  in  the  above  rule,  of  the 
aberrations  and  nutations  of  the  star,  nor  of  some  other  small  cor¬ 
rections  of  the  elements,  as  they  would  produce  but  little  effect 
on  the  results  obtained  by  the  projection. 

2.  The  calculation  of  the  true  times  of  immersion  and  emersion, 
from  the  approximate  times,  may  be  made  nearly  in  the  same 
manner  as  for  the  beginning  and  end  of  an  eclipse  of  the  sun. 
There  are,  however,  the  following  differences  in  the  calculation: 
The  approximate  times  of  immersion  and  emersion  must  be  used 
instead  of  the  approximate  times  of  beginning  and  greatest  ob¬ 
scuration,  or  greatest  obscuration  and  end  of  the  eclipse.  The 
star’s  longitude,  corrected  for  aberration  and  nutation,  must  be 
used  instead  of  the  sun’s  longitudes.  The  apparent  distances  of 
the  moon  from  the  star  in  latitude,  must  be  used  instead  of  the 
moon’s  latitudes.  To  the  logarithm  of  G,  obtained  from  the  moon 
and  star’s  longitudes,  add  the  Cosine  of  the  star’s  latitude,  reject¬ 
ing  the  tens  in  the  index,  and  use  the  natural  number  correspond¬ 
ing  to  the  sum,  instead  of  G.  To  the  logarithm  of  M,  add  the 
Cosine  of  the  star’s  latitude,  rejecting  the  tens  in  the  index,  and 
use  the  natural  number  corresponding  to  the  sum,  instead  of  M. 
Lastly,  the  moon’s  augmented  semidiameter  must  be  used,  instead 
©f  the  sum  of  the  semidiameters  of  the  sun  and  moon. 

3.  The  projection  or  calculation  of  an  occupation  of  a  Planet 
by  the  moon,  may  be  performed  in  nearly  the  same  manner  as 
for  a  fixed  star.  The  planet’s  right  ascension,  declination,  geo¬ 
centric  longitude  and  latitude,  may  be  obtained  from  the  Nautical 
Almanac,  and  must  be  used  instead  of  those  of  the  star.  The 
moon’s  hourly  motions  from  the  planet  in  longitude  and  latitude, 
must  be  used  instead  of  the  hourly  motions  of  the  moon.  When 
great  accuracy  is  required,  the  parallax  and  semidiameter  of  the 
planet  must  be  taken  into  view;  but  it  is  not  thought  necessary  to 
notice  here  the  manner  of  doing  this. 

Exam.  Required  to  project,  for  the  latitude  and  meridian  of 
Washington,  an  occupation  of  y  Tauri ,  by  the  moon,  which  took 
place  in  January,  1813,  the  elements  obtained  from  the  Nautical 

46 


3  5% 


ASTRONOMY. 


Almanac  for  that  year,  and  by  the  problems  referred  to  in  the  rule, 
being  as  follows: 


63 
292  30 


Conjunction,  in  appar.  time  at  Washington,  January, 
Star’s  passage  over  meridian,  - 
Semidiameter  of  circle  of  projection, 

Star’s  longitude,  *  - 
Sun’s  longitude, 

Star’s  latitude,  ... 

Moon’s  latitude,  - 

Moon’s  dist.  from  star  in  latitude,  north , 

Star’s  declination,  north , 

Moon’s  hor.  mot.  in  longitude, 

Moon’s  hor.  mot.  in  latitude,  tending  south , 

Moon’s  semidiameter,  - 
Latitude  of  Place,  ... 

Fourth  term,  - 


d.  h.  m. 
12  6  53 
8  32 

59'  29"  =  59'.48 

11 


o 

5 

15 


5  S. 

55  S. 

10  =  40.17 


45 
4 

40 
10 

35  57  =  35.95 
0  45  =  0.75 
16  14  =  16.23 


38  53 

31 


45  =  31.75 


The  Figures  61,  62,  63  and  64,  are  adapted  to  this  example, 
and  need  no  further  explanation.  The  time  of  immersion,  ob¬ 
tained  by  the  construction,  is  5h.  49  m.;  and  of  the  emersion, 
6  h.  46  m. 


PROBLEM  XXXIX. 

Given  the  Moon's  true  Longitude  to  find  the  corresponding  time 
at  Greenwich  by  the  Nautical  Almanac ,  the  approximate  time  be - 
ing  given. 

Call  the  hours  and  minutes,  &c.  of  the  approximate  time,  or 
their  excess  above  12  hours,  T. 

Take,  from  the  Almanac,  the  two  longitudes  of  the  moon  next 
less,  and  two  next  greater  than  the  given  longitude,  and  find  the 
first  and  second  differences,  and  the  arcs  A  and  B,  as  directed  in 
prob.  XII.  With  the  time  T  and  the  arc  B,  take  the  equation  of 
seepnd  differences  from  table  LYI,  and  apply  it  with  the  same 
sign  as  B,  to  the  difference  between  the  second  longitude  and  the 
given  one,  and  call  the  result  D.  Then  A  :  D  : :  1 2  hours  :  to 


ASTRONOMY. 


a  55 


the  required  time,  reckoned  from  the  noon  or  midnight  corres¬ 
ponding  to  the  second  longitude.  The  time  thus  obtained  will  be 
apparent  time  at  Greenwich. 


Exam.  Required  the  time  at  Greenwich  when  the  moon’s  longi¬ 
tude  is  7*  18°  58'  47",  the  approximate  time  being  August  6th, 
1821,  at  1  h.  41  m. 

Here,  T  =  1  h.  41  m. 


5th  midn. 
6th  noon 
6th  midn. 
7th  noon 


Longitudes. 

1st  Diff. 

2d  Diff. 

Mean  of 

2d  Diff. 

7*  12°  8'  55" 
7  18  7  55 

7  24  9  18 

8  0  13  38 

5° 59'  0" 
A.  6  1  23 
|  6  4  20 

2  23 

2  57 

B.  +  2'  40'' 

Given  longitude,  -  -  7s  18°  58'  47" 

2d  longitude,  -  -  -  7187  55 


Difference,  -  50  52 

Equat.  2d  Diff.  -  -  -  -J-  9.7 

D.  51  1.7 


6°  1'  23"  :  51'  1".7  : :  12  h.  :  1  h.  41  m.  40  sec.  the  time  re¬ 
quired. 


PROBLEM  XL. 

Given  the  Latitude  of  a  Place  and  the  observed  apparent  Time 
of  the  Beginning  or  End  of  an  Occultation  of  a  fixed  star  by  the 
moony  to  find  the  Longitude  of  the  place ,  it  being  supposed  to  be 
nearly  known  by  estimation. 

By  means  of  the  estimated  longitude,  reduce  the  observed  time 
to  the  meridian  of  Greenwich,  and  for  that  time  calculate  the 
moon’s  true  longitude,  latitude,  and  semidiameter;  and  then  the 
parallax  in  longitude,  the  apparent  latitude,  and  the  augmented 
semidiameter.  Also,  find  the  star’s  longitude  and  latitude,  cor¬ 
rected  for  aberration  and  nutation.  When  the  moon’s  apparent 
latitude  and  the  latitude  of  the  star  are  of  the  same  name,  take 


ASTRONOMY. 


S5C) 

their  difference;  but  when  they  are  of  different  names,  take  tlieir 
sum;  the  result  will  be  the  moon’s  apparent  distance  from  the  star 
in  latitude.  Call  this  distance  d,  and  the  augmented  semidiame¬ 
ter  s.  Add  together  the  logarithms  of  (s  4-  d)  and  (s  —  d),  and 
to  half  their  sum,  add  the  arithmetical  complement  of  the  Cosine 
of  the  stars  latitude,  and  the  result  will  be  the  logarithm  of  a 
small  arc  c.  When  the  calculation  is  for  the  beginning ,  subtract  c 
from  the  star’s  longitude:  but  when  it  is  for  the  end ,  add  c  to  the 
star’s  longitude;  and  the  result  will  be  the  moon’s  apparent  longi¬ 
tude  at  the  observed  time  of  beginning  or  end.  To  the  moon’s 
apparent  longitude,  thus  found,  apply  the  parallax  in  longitude, 
by  adding  when  the  moon  is  to  the  west  of  the  nonagesimal,  but 
by  subtracting  when  it  is  to  the  east;  and  the  result  will  be  the 
moon’s  true  longitude,  as  deduced  from  the  observation.  Find 
from  the  Nautical  Almanac,  by  the  last  problem,  the  time  at 
Greenwich  when  the  moon  has  this  longitude,  the  approximate 
time  being  the  observed  time  of  beginning  or  end  reduced  to  the 
meridian  of  Greenwich.  Then,  on  the  supposition  that  the  tables 
are  accurate,  the  difference  between  the  time  found  from  the 
Nautical  Almanac,  and  the  observed  time  of  beginning  or  end, 
will  be  the  longitude  of  the  place  in  time.  If  the  longitude  thus 
found  differs  considerably  from  the  estimated  longitude,  the  ope¬ 
ration  should  be  repeated. 

Note  1.  When  the  immersion  and  emersion  are  both  observed, 
the  longitude  should  be  deduced  from  each,  and  the  mean  of  the 
results  taken  as  the  longitude  of  the  place. 

2.  The  above  rule  with  a  little  change,  will  serve  to  determine 
the  longitude  of  a  place  from  the  observed  time  of  beginning  or 
end  of  an  eclipse  of  the  sun.  To  do  this,  the  sun’s  longitude  must 
be  used  instead  of  the  star’s;  d  must  be  taken  equal  to  the  moon’s 
apparent  latitude,  and  s  equal  to  the  sum  of  the  sun’s  semidiame¬ 
ter  and  the  augmented  semidiameter  of  the  moon.  It  may  also  be 
observed,  that  the  sun’s  latitude  being  nothing,  the  arithmetical 
complement  of  its  Cosine  will  be  nothing. 

Exam.  The  beginning  of  the  occultation  of  y  Tauri ,  mentioned 
in  the  example  to  prob.  XXXVIII,  was  observed  by  Bradley  and 


ASTRONOMY. 


3  57 

Pease,  at  a  distance  of  nearly  two  miles  from  the  Capitol  in 
Washington.  The  apparent  time  of  immersion,  after  allowance 
made  for  the  error  of  the  watch,  was  5  h.  46  m.  49  sec.;  the  re¬ 
duced  latitude  of  the  place  of  observation,  38°  42'  59"  N. ;  and 
its  estimated  longitude  in  time,  5  h.  7  m.  50  sec.  west.  Required 
the  longitude  of  the  place  of  observation,  making  use  of  the  fol¬ 
lowing  elements,  obtained  from  the  Nautical  Almanac  for  1813, 
or  calculated  by  preceding  problems: 


Star’s  corrected  longitude, 

Do.  latitude,  south, 

Moon’s  parallax  in  longitude,  the  moon  being 
to  the  east  of  the  nonagesimal, 


63° 

5 


11' 

45 


18".2 

6.1 


24  59.8 


Moon’s  apparent  latitude,  south, 

- 

5  37  37.1 

Moon’s  appar.  dist.  from  star  in  latitude, 

d.  7  29 

Moon’s  augmented  semidiameter, 

16  23.6 

s  +  d  -  -  1432".  6 

log.  3.15613 

s  —  d  -  -  -  534.6 

log.  2.72803 

2)5.88416 

2.94208 

Star’s  latitude,  5°  45'  Ar.  Co. 

cos.  0.00219 

c  -  -  879". 6  e=  14'  39".6 

log.  2.94427 

Star’s  longitude,  - 

63° 

11'  1 8".2 

c . 

14  39.6 

Moon’s  apparent  longitude, 

62 

56  38.6 

Parallax  in  longitude, 

24  59.8 

Moon’s  true  longitude, 

62 

31  38.8 

Appar.  time  at  Greenwich  when  the  moon 

h.  m.  sec. 

had  that  longitude, 

10  54  39.4 

Appar.  time  of  immersion,  observed, 

- 

5  46  49 

Longitude,  in  time,  of  the  place  of  observation,  5  7  50.4 


END  OF  PART  II. 


, 


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ASTRONOMICAL 


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TABLE  I 


3 


Latitudes ,  and  Longitudes  from  the  Meridian  of  Greenwich ,  of 
some  Cities ,  and  other  conspicuous  Places. 


Names  of  Places. 

Latitude. 

Longitude  in 
Degrees. 

Longitude 
in  Time. 

o  in 

h.  m.  s. 

Amsterdam, 

Holland, 

52  22  17N. 

4  53  15E. 

0  19  33 

Athens, 

Greece, 

37  58  IN. 

23  46  14E. 

1  35  5 

Baltimore, 

U.  States, 

39  23  ON. 

76  50  0W. 

5  7  20 

Bergen, 

Norway, 

60  24  ON. 

5  20  40E. 

0  21  23 

Berlin, 

Germany, 

52  31  45N. 

13  22  15E. 

0  53  29 

Boston, 

U.  States, 

42  23  ON. 

71  4  0W. 

4  44  16 

Botany  Bay, 

New  Hoi. 

34  3  OS. 

151  15  0E. 

10  5  0 

Brest, 

France, 

48  23  14N. 

4  28  45W. 

0  17  55 

Bristol, 

England, 

51  27  6N. 

2  35  29W. 

0  10  22 

Cadiz,  Obs. 

Spain, 

36  32  ON. 

6  17  22 W. 

0  25  9 

Cairo, 

Egypt, 

30  2  21N. 

31  18  45E. 

2  5  15 

Canton, 

China, 

23  8  9N. 

113  2  45E. 

7  32  11 

Cape  G.  Hope, 

Africa, 

33  55  15S. 

18  24  0E. 

1  15  36 

Charleston. 

U.  States, 

32  50  ON. 

79  48  0W. 

5  19  12 

Constantinople, 

Turkey, 

41  1  27N. 

28  55  15E. 

1  55  41 

Copenhagen, 

Denmark, 

55  41  4N. 

12  35  6E. 

0  50  20 

Dublin, 

Ireland, 

53  21  11N. 

6  18  45W. 

0  25  15 

Edinburgh, 

Scotland, 

55  57  57N. 

3  10  15W. 

0  12  41 

Greenwich,  Obs. 

England, 

51  28  40N. 

0  0  0 

0  0  0 

London, 

England, 

51  30  49N. 

0  5  30W. 

0  0  22 

Madrid, 

Spain, 

40  24  57N. 

3  42  15 W. 

0  14  49 

Naples, 

Italy, 

40  50  15N. 

14  15  45 E. 

0  57  3 

New-Haven, 

U.  States, 

41  18  ON. 

72  58  0W. 

4  51  52 

New-York, 

U.  States, 

40  42  40N. 

74  1  OW. 

4  56  4 

Paris,  Obs. 

France, 

48  50  14N. 

2  20  15E. 

0  9  21 

Pekin, 

China, 

39  54  13N. 

116  27  45E. 

7  45  51 

St.  Petersburg, 

Russia, 

59  56  23N. 

30  18  45E. 

2  1  15 

Philadelphia, 

U.  States, 

39  56  55 N. 

75  11  30W. 

5  0  46 

Point  Venus, 

Otaheite, 

17  29  17S. 

149  30  15W. 

9  58  1 

Quebec, 

Canada, 

46  47  30N. 

71  9  45 W. 

4  44  39 

Richmond, 

U.  States, 

37  30  ON. 

77  58  OW. 

5  11  52 

Rome, 

Italy, 

41  53  54N. 

12  28  15E. 

0  49  53 

Stockholm, 

Sweden, 

59  20  31N. 

18  3  30E. 

1  12  14 

Vienna, 

Germany, 

48  12  40N. 

16  22  45E. 

1  5  31 

Wardhus, 

Lapland, 

70  22  36N. 

26  26  SOW. 

1  45  46 

Washington, 

U.  States, 

38  53  ON. 

76  55  30 W. 

5  7  42 

4 


TABLE  II 


Mean  Astronomical  Refractions , 


Ap. 

Alt. 

Refr. 

Ap. 

Alt. 

Refr. 

Ap. 

Alt. 

Refr. 

Ap. 

Alt. 

Refr.  1 

0°  0' 

33' 

0" 

4°  0' 

IT  51" 

12°  20' 

4' 

16" 

45°  0' 

O'  57" 

0 

5 

32 

10 

4 

10 

11 

29 

12 

40 

4 

9 

46 

0 

0 

55 

0 

10 

31 

22 

4 

20 

11 

8 

13 

0 

4 

3 

47 

0 

0 

53 

0 

15 

30  35 

4 

30 

10 

48 

13 

20 

3 

57 

48 

0 

0 

51 

0 

20 

29 

50 

4 

40 

10 

29 

13 

40 

3 

51 

49 

0 

0 

49 

0 

25 

29 

6 

4 

50 

10 

11 

14 

0 

3 

45 

50  0 

0 

48 

0 

30 

28 

22 

5 

0 

9 

54 

14 

20 

3 

40 

51 

0 

0 

46 

0 

35 

27 

41 

5 

10 

9 

38 

14 

40 

3 

35 

52 

0 

0 

44 

0 

40 

27 

0 

5 

20 

9 

23 

15 

0 

3 

30 

53 

0 

0 

43 

0 

45 

26 

20 

5 

30 

9 

8 

15 

30 

3 

24 

54 

0 

0 

41 

0 

50 

25 

42 

5 

40 

8 

54 

16 

0 

3 

17 

55 

0 

0 

40 

0 

55 

25 

5 

5 

50 

8 

41 

16 

30 

3 

10 

56 

0 

0  38 

1 

0 

24 

29 

6 

0 

8 

28 

17 

0 

3 

4 

57  0 

0 

37 

1 

5 

23 

54 

6 

10 

8 

15 

17 

30 

2 

59 

58 

0 

0  35 

1 

10 

23 

20 

6 

20 

8 

3 

18 

0 

2 

54 

59 

0 

0 

34 

1 

15 

22 

47 

6 

30 

7 

51 

18 

30 

2 

49 

60 

0 

0 

33 

1 

20 

22 

15 

6 

40 

7 

40 

19 

0 

2 

45 

61 

0 

0 

31 

1 

25 

21 

44 

6 

50 

7 

30 

19 

30 

2 

39 

62 

0 

0 

30 

1 

30 

21 

15 

7 

0 

7 

20 

20 

0 

2 

35 

63 

0 

0 

29 

1 

35 

20 

46 

7 

10 

7 

11 

20 

30 

2 

31 

64 

0 

0 

28 

1 

40 

20 

18 

7 

20 

7 

2 

21 

0 

2 

27 

65 

0 

0 

26 

1 

45 

19 

51 

7  30 

6 

53 

21 

30 

2 

24 

66 

0 

0 

25 

1 

50 

19 

25 

7 

40 

6 

45 

22 

0 

2 

20 

67  0 

0 

24 

1 

55 

19 

0 

7 

50 

6 

37 

23 

0 

2 

14 

68 

0 

0  23 

2 

0 

18 

35 

8 

0 

6 

29 

24 

0 

2 

7 

69 

0 

0 

22 

2 

5 

18 

11 

8 

10 

6 

22 

25 

0 

2 

2 

70 

0 

0 

21 

2 

10 

17 

48 

8 

20 

6 

15 

26 

0 

1 

56 

71 

0 

0 

19 

2 

15 

17 

26 

8 

30 

6 

8 

27 

0 

1 

51 

72 

0 

0 

18 

2 

20 

17 

4 

8 

40 

6 

1 

28 

0 

1 

47 

73 

0 

0 

17 

2 

25 

16  44 

8 

50 

5 

55 

29 

0 

1 

42 

74 

0 

0 

16 

2 

30 

16 

24 

9 

0 

5 

48 

30 

0 

1 

38 

75 

0 

0 

15 

2 

35 

16 

4 

9 

10 

5 

42 

31 

0 

1 

35 

76  0 

0 

14 

2 

40 

15 

45 

9 

20 

5 

36 

32 

0 

1 

31 

77 

0 

0 

13 

2 

45 

15 

27 

9 

30 

5 

31 

33 

0 

1 

28 

78 

0 

0 

12 

2 

50 

15 

9 

9 

40 

5 

25 

34 

0 

1 

24 

79  0 

0 

11 

2 

55 

14  52 

9 

50 

5 

20 

35 

0 

1 

21 

80 

0 

0 

10 

3 

0 

14 

36 

10 

0 

5 

15 

36 

0 

1 

18 

81 

0 

0 

9 

3 

5 

14 

20 

10 

15 

5 

7 

37 

0 

1 

16 

82 

0 

0 

8 

3 

10 

14 

4 

10  30 

5 

0 

38 

0 

1 

13 

83 

0 

0 

7 

3 

15 

13 

49 

10  45 

4 

53 

39 

0 

1 

10 

84 

0 

0 

6 

3 

20 

13 

34 

11 

0 

4 

47 

40 

0 

1 

8 

85 

0 

0 

5 

3 

25 

13 

20 

11 

15 

4 

40 

41 

0 

1 

5 

86 

0 

0 

4 

3 

30 

13 

6 

11 

30 

4  34 

42 

0 

1 

3 

87 

0 

0 

3 

3 

40 

12  40 

11 

45 

4 

29 

43 

0 

1 

1 

88 

0 

0 

2 

3 

50 

12 

15 

12 

0 

4 

23 

44 

0 

0 

59 

89 

0 

0 

1 

TABLE  III, 


5 


Mean  Right  Ascensions  and  Declinations  of  some  of  the  Fixed 
Stars,  for  the  beginning  of  1820,  with  their  Annual  Variations. 


Names  and  Magnitude. 

Right  Asc. 

An.  Var. 

Declination. 

An.  Var. 

y  Pegasi,  -  - 

Mag. 

o 

o  ,  n 

0  59  35 

// 

+  46.1 

o  ,  n 

14  10  56N. 

t 

+  20.0 

«  Polaris,  -  - 

2.3 

14  13  7 

216.4 

88  20  55N. 

+  19.4 

a  Arietis,  -  - 

-  3 

29  15  38 

50.3 

22  36  23N. 

+ 17 .3 

a.  Ceti,  -  -  - 

2 

43  13  8 

46.7 

3  22  39N. 

-t-14.5 

y  Tauri,  -  - 

.  2  • 

62  23  23 

51.0 

15  11  3N. 

+  9.2 

Aldebaran,  - 

-  1 

66  24  0 

51.4 

16  8  19N. 

+  7.8 

Capella, 

-  1 

75  51  7 

66.3 

45  48  8N. 

+  4.5 

Rigel,  -  - 

-  1 

76  28  21 

51.8 

8  25  2S. 

—  4.7 

Tauri,  -  - 

-  2 

78  43  47 

56.7 

28  26  42N. 

+  3.8 

£  Tauri,  -  - 

-  3 

81  43  14 

53.6 

21  1  23N. 

+  2.8 

»  Gemi  norum, 

2.3 

91  0  6 

54.3 

22  32  56N. 

—  0.4 

fx  Geminorum, 

-  3 

93  1  0 

54.5 

22  35  48N. 

—  1.1 

y  Geminorum, 

2.3 

96  49  37 

52.0 

16  32  36N. 

—  2.5 

Sirius,  -  - 

-  1 

99  18  18 

39.8 

16  28  33S. 

+  4.4 

Geminorum, 

-  3 

107  20  25 

53.8 

22  18  15N. 

—  6.0 

Procyon,  -  - 

1.2 

112  28  2 

47.1 

5  40  46N. 

—  8.6 

Pollux,  -  - 

2.3 

113  34  16 

55.3 

28  27  7N. 

—  8.0 

u  Leonis,  -  - 

-  O 

149  22  33 

49.2 

17  38  10N. 

—17.3 

Regulus,  -* 
Virginia,  -  - 

-  1 

149  41  39 

48.1 

12  50  36N. 

—17.3 

-  3 

175  19  45 

46.9 

2  46  44N. 

—20.3 

y  Virginia,  - 

-  3 

188  8  4 

45.3 

0  27  38S. 

+  20.0 

et  Virginis,  -  - 

-  1 

198  55  30 

47.2 

10  13  5S. 

+  19.0 

A  returns,  - 

-  1 

211  51  45 

40.9 

20  7  28N. 

—19.0 

tt  2  Librae,  -  - 

2.3 

220  14  2 

49.5 

15  17  13S. 

+  15.4 

cT  Scorpii, 

/3  Scorpii,  -  - 

-  3 

237  25  37 

52.9 

22  5  59S. 

+  10.9 

-  2 

238  44  48 

52.0 

19  18  13S. 

+  10.5 

Antares,  - 

-  1 

244  35  49 

54.9 

26  1  21S. 

+  8.7 

ct  Lyra;,  -  -  - 

-  1 

277  42  37 

30.4 

38  37  19N. 

4-  2.9 

<r  Sagittarii,  - 

-2.3 

281  1  28 

55.8 

26  30  34S. 

—  3.7 

7r  Sagittarii,  - 

-  3 

284  45  47 

53.5 

21  18  OS. 

—  5.0 

at  1  Capricorni, 

3.4 

301  54  56 

50.0 

13  3  22S* 

—10.5 

et  2  Capricorni, 
(S  Capricorni, 

-  3 

302  0  53 

50.0 

13  5  33S. 

—10.8 

-  3 

302  43  18 

50.6 

15  20  29S. 

—10.8 

V  Capricorni,  - 

3.4 

322  31  32 

50.0 

17  28  10S. 

—15.8 

J  Capricorni, 

-  3 

324  16  24 

49.8 

16  56  16S. 

—16.0 

Fomalhaut,  - 

-  1 

341  55  14 

50.1 

30  34  25S. 

—18.8 

6 


TABLE  IV, 


Mean  New  Moons  and  Arguments ,  in  January. 


Mean  New 
Moon  in 
January. 

I. 

H. 

III. 

IV. 

N. 

A.D. 

1821 

D.  H.  M. 
2  17  59 

0092 

7859 

80 

78 

823 

1822 

21  15  32 

0602 

7182 

78 

66 

930 

1823 

11  0  20 

0304 

5787 

61 

55 

953 

1824B. 

29  21  53 

0814 

5110 

59 

43 

060 

1825 

18  6  41 

0516 

3716 

42 

32 

083 

1826 

7  15  30 

0218 

2321 

25 

21 

105 

1827 

26  13  3 

0728 

1644 

24 

09 

213 

1828  B. 

15  21  51 

0430 

0250 

07 

98 

235 

1829 

4  6  40 

0131 

8855 

90 

87 

257 

1830 

23  4  12 

0642 

8178 

88 

75 

365 

1831 

12  13  1 

0343 

6784 

71 

64 

387 

1832  B. 

1  21  50 

0045 

5389 

54 

53 

409 

1833 

19  19  22 

0555 

4712 

53 

42 

517 

1834 

9  4  11 

0257 

3318 

36 

31 

539 

1835 

28  1  43 

0768 

2641 

34 

19 

647 

1836  B. 

17  10  32 

0469 

1246 

17 

08 

669 

1837 

5  19  20 

0171 

9852 

00 

97 

692 

1838 

24  16  53 

0681 

9175 

99 

8* 

799 

1839 

14  1  42 

0383 

7780 

82 

74 

822 

1840  B. 

3  10  30 

0085 

6386 

65 

63 

844 

1841 

21  8  3 

0595 

5709 

63 

51 

951 

1842 

10  16  51 

0297 

4314 

46 

40 

974 

1843 

29  14  24 

0807 

3637 

44 

28 

081 

1844  B. 

18  23  13 

0509 

2243 

28 

17 

104 

1845 

7  8  1 

0211 

0848 

11 

06 

126 

1846 

26  5  34 

0721 

0171 

09 

94 

234 

1847 

15  14  22 

0423 

8777 

92 

84 

256 

1848  B. 

4  23  11 

0125 

7382 

75 

73 

278 

1849 

22  20  43 

0635 

6705 

73 

61 

386 

1850 

12  5  32 

0337 

5311 

56 

50 

408 

1851 

1  14  21 

0038 

3916 

40 

39 

431 

1852  B. 

20  11  53 

0549 

3239 

38 

27 

538 

1853 

8  20  42 

0251 

1845 

21 

16 

560 

1854 

27  18  14 

0761 

1168 

19 

04 

668 

1855 

17  3  3 

0463 

9773 

02 

93 

690 

1856  B. 

6  11  51 

0164 

8379 

85 

82 

713 

1857 

24  9  24 

0675 

7702 

84 

70 

820 

1858 

13  18  13 

0376 

6307 

67 

59 

843 

1859 

3  3  1 

0078 

4913 

50 

48 

865 

1860  B. 

22  0  34 

0588 

4236 

48 

36 

972 

TABLE  V. 


7 


Mean  Lunations  and  Change  in  Argument*. 


Num 

Lunations. 

I. 

II. 

IIL 

IV. 

N. 

$ 

D.  H.  M. 

14  18  22 

404 

5359 

58 

50 

43 

1 

29  12  44 

808 

717 

15 

99 

85 

2 

59  1  28 

1617 

1434 

31 

98 

170 

3 

88  14  12 

2425 

2151 

46 

97 

256 

4 

118  2  56 

3234 

2869 

61 

96 

341 

5 

147  15  40 

4042 

3586 

76 

95 

426 

6 

177  4  24 

4851 

4303 

92 

95 

511 

7 

206  17  8 

5659 

5020 

7 

94 

596 

-  8 

236  5  52 

6468 

5737 

22 

93 

682 

9 

265  18  36 

7276 

6454 

37 

92 

767 

10 

295  7  20 

8085 

7171 

53 

91 

852 

11 

324  20  5 

8893 

7889 

68 

90 

937 

12 

354  8  49 

9702 

8606 

83 

89 

22 

13 

383  21  33 

510 

9323 

98 

88 

108 

TABLE  VI. 

Number  of  Days  from  the  commencement  of  the  'gear 
to  the  first  of  each  month . 


Months. 

Com. 

Bis. 

January,  - 

Days. 

0 

Days. 

0 

February  - 

31 

31 

March,  -  - 

59 

60 

April,  -  - 

90 

91 

May,  -  - 

120 

121 

June,  -  -  - 

151 

152 

July,  -  - 

181 

182 

August,  -  . 

212 

213 

September, 

243 

244 

October,  - 

273 

274 

November, 

304 

305 

December, 

334 

335 

8 


TABLE  VII, 


Equations  for  New  and  Full  Mom . 


Arg. 

L 

11. 

Arg. 

»• 

II 

Yrg. 

111. 

iV. 

Arg 

H.M. 

H.  M. 

H.  M. 

H.  Vf. 

M 

M. 

000 

4  20 

10  10 

5000 

4  20 

10  10 

25 

3 

31 

25 

100 

4  36 

9  36 

5100 

4  5 

10  50 

26 

3 

31 

24 

200 

4  52 

9  2 

5200 

3  49 

11  30 

27 

3 

30 

23 

300 

5  8 

8  28 

5300 

3  34 

12  9 

28 

3 

30 

22 

400 

5  24 

7  55 

5400 

3  19 

12  48 

29 

3 

30 

21 

500 

5  40 

7  22 

5500 

3  4 

13  26 

30 

3 

30 

20 

600 

5  55 

6  49 

5600 

2  49 

14  3 

31 

3 

30 

19 

700 

6  10 

6  17 

5700 

2  35 

14  39 

32 

4 

30 

18 

800 

6  24 

5  46 

5800 

2  21 

15  13 

33 

4 

29 

17 

900 

6  38 

5  15 

5900 

2  8 

15  46 

34 

4 

29 

16 

1000 

6  51 

4  46 

6000 

1  55 

16  18 

35 

4 

29 

15 

1100 

7  4 

4  17 

6100 

1  42 

16  48 

36 

5 

28 

14 

1200 

7  15 

3  50 

6200 

1  31 

17  16 

37 

5 

28 

13 

1300 

7  27 

3  24 

6300 

1  19 

17  42 

38 

5 

27 

12 

1400 

7  37 

2  59 

6400 

1  9 

18  6 

39 

5 

27 

11 

1500 

7  47 

2  35 

6500 

0  59 

18  28 

40 

6 

26 

10 

1600 

7  55 

2  14 

6600 

0  50 

18  48 

41 

6 

26 

9 

1700 

8  3 

1  53 

6700 

0  42 

19  6 

42 

7 

25 

8 

1800 

8  10 

1  35 

6800 

0  34 

19  21 

43  : 

7 

25 

7 

1900 

8  16 

1  18 

6900 

0  28 

19  33 

44 

7 

24 

6 

2000 

8  21 

1  3 

7000 

0  22 

19  44 

45 

8 

23 

5 

2100 

8  25 

0  51 

7100 

0  17 

19  52 

46 

8 

23 

4 

2200 

8  29 

0  40 

7200 

0  14 

19  57 

47 

9 

22 

3 

2300 

8  31 

0  32 

7300 

0  11 

20  0 

48 

9 

21 

2 

2400 

8  32 

0  25 

7400 

0  9 

20  1 

49 

10 

21 

1 

2500 

8  32 

0  21 

7500 

0  8 

19  59 

50 

10 

20 

0 

2600 

8  31 

0  19 

7600 

0  8 

19  55 

51 

10 

19 

99 

2700 

8  29 

0  20 

7700 

0  9 

19  48 

52 

11 

19 

98 

2800 

8  26 

0  23 

7800 

0  11 

19  40 

53 

11 

18 

97 

2900 

8  23 

0  28 

7900 

0  15 

19  29 

54 

12 

17 

96 

3000 

8  18 

0  36 

8000 

0  19 

19  17 

55 

12 

17 

95 

3100 

8  12 

0  47 

8100 

0  24 

19  2 

56 

13 

16 

94 

3200 

8  6 

0  59 

8200 

0  30 

18  45 

57 

13 

15 

93 

3300 

7  58 

1  14 

8300 

0  37 

18  27 

58 

13 

15 

92 

3400 

7  50 

1  32 

8400 

0  45 

18  6 

59 

14 

14 

91 

3500 

7  41 

1  52 

8500 

0  53 

17  45 

60 

14 

14 

90 

3600 

7  31 

2  14 

8600 

1  3 

17  21 

61 

15 

13 

89 

3700 

7  21 

2  38 

8700 

1  13 

16  56 

62 

15 

13 

88 

3800 

7  9 

3  4 

8800 

1  25 

16  30 

63 

15 

12 

87 

3900 

6  58 

3  32 

8900 

1  36 

16  3 

64 

15 

12 

86 

4000 

6  45 

4  2 

9000 

1  49 

15  34 

65 

16 

11 

85 

4100 

6  32 

4  34 

9100 

2  2 

15  5 

66 

16 

11 

84 

4200 

6  W 

5  7 

9200 

2  16 

14  34 

67 

16 

11 

83 

4300 

6  5 

5  41 

9300 

2  30 

14  3 

68 

16 

10 

82 

4400 

5  51 

6  17 

9400 

2  45 

13  31 

69 

17 

10 

81 

4500 

5  36 

6  54 

9500 

3  0 

12  58 

70 

17 

10 

80 

4600 

5  21 

7  32 

9600 

3  16 

12  25 

71 

17 

10 

79 

4700 

5  6 

8  11 

9700 

3  32 

11  52 

72 

17 

10 

78 

4800 

4  51 

8  50 

9800  , 

3  48 

11  18 

73 

17 

10 

77 

4900 

4  35 

9  30 

9900 

4  4 

10  44 

74 

17 

9 

76 

5000 

4  20 

10  10 

10000  1 

4  20 

10  10 

75 

17 

9 

75 

TABLE  VIII 


9 


Mean  Longitudes  and  Latitudes  of  some  of  the  Ficced  Stars,  for 
the  beginning  of  1810,  with  their  Annual  Variations . 


Names  and  Magi 

itudes 

Longitude. 

An.Var. 

Latitude. 

An.  Var. 

Mag. 

£ 

o 

r  n 

// 

o  ,  n 

it 

y  Pegasi,  -  - 
a  Arietis,  -  - 

2 

0 

6 

30  27 

4-  50.09 

12  35  41N. 

+  0.12 

-  3 

1 

5 

0  21 

50.27 

9  57  38N. 

+  0.18 

y  Tauri,  -  - 

.  2 

2 

3 

8  30 

50.21 

5  45  6S. 

—  0.45 

e  Tauri,  -  - 

3.4 

2 

5 

48  11 

50.20 

2  35  10S. 

—  0.46 

Aldebaran,  - 

-  1 

2 

7 

8  1 

50.20 

5  28  49S. 

—  0.32 

Bigel,  -  - 

-  1 

2 

14 

10  23 

50.24 

31  .8  48S. 

—  0.47 

Capella, 
Tauri,  -  - 

-  1 

2 

19 

12  03 

50.19 

22  52  7N. 

+  0.48 

-  2 

2 

19 

55  14 

50.20 

5  22  21N. 

-f"  0.48 

£  Tauri,  -  - 

-  3 

2 

22 

7  48 

50.20 

2  13  5S. 

—  0.48 

a  Geminorum, 

2.3 

3 

0 

47  10 

50.20 

0  54  38S. 

r-  0.48 

y.  Geminorum, 

-  3 

3 

2 

38  30 

50.20 

0  50  8S. 

—  0.47 

y  Geminorum, 

2.3 

3 

6 

26  51 

.50.18 

6  45  45S. 

—  0.47 

e  Geminorum, 

-  3 

3 

7 

17  2 

50.20 

2  2  51N. 

-f-  0.46 

Sirius,  -  - 

-  1 

3 

11 

28  13 

50.07 

39  22  36S. 

—  0.45 

S  Geminorum, 

-  3 

3 

15 

51  59 

50.20 

0  11  59S. 

—  0.44 

Pollux,  -  - 

2.3 

3 

20 

35  34 

49.47 

6  40  16N. 

4.  0.28 

a  Leonis,  -  - 

-  o 

4 

25 

14  55 

50.23 

4  51  17N. 

+  0.22 

Regulus,  - 

-  1 

4 

27 

11  18 

50.00 

0  27  36N. 

4-  0.20 

jS  Virginis,  -  - 

-  3 

5 

24 

27  32 

50.20 

0  41  32N. 

—  0.02 

a  Virginis,  - 

4.3 

6 

2 

10  58 

50.21 

1  22  24N. 

—  0.08 

y  Virginis,  - 

-  3 

6 

7 

31  20 

50.22 

2  48  45N. 

—  0.13 

a.  Virginis,  -  - 

-  1 

6 

21 

11  22 

50.06 

2  2  14S. 

4.  0.08 

^  Arcturus,  - 

-  1 

6 

21 

35  5 

50.45 

30  54  3N. 

—  0.24 

2  Librae,  -  - 

2.3 

7 

12 

26  4 

50.20 

0  21  32N. 

—  0.37 

7  Lybrae,  -  - 

3.4 

7 

22 

28  43 

50.22 

4  24  28N. 

—  0.42 

S  Scorpii, 

-  3 

7 

29 

55  0 

50.19 

1  57  33 

4-  0.44 

it  Scorpii, 

-  3 

8 

0 

17  5 

50.18 

5  26  54S. 

-f  0.45 

Scorpii,  -  - 

-  2 

8 

0 

32  5 

50.20 

1  2  2N. 

—  0.45 

Antares, 

-  1 

8 

7 

6  34 

50.14 

4  32  22S. 

-f  0.17 

a-  Sagittarii,  -  . 

.2.3 

9 

9 

43  47 

50.21 

3  25  14S. 

4-  0.46 

a.  Lyrae,  -  - 

-  1 

9 

12 

39  0 

49.89 

61  44  31N. 

—  0.45 

7r  Sagittarii,  - 

-  3 

9 

13 

35  56 

50.19 

1  27  51N. 

— .  0.45 

/g  Capricorni, 

-  3 

10 

1 

23  31 

50.17 

4  36  35N. 

—  0.37 

7  Capricorni,  - 

3.4 

10 

19 

7  41 

50.21 

2  32  13S. 

4-  0.26 

S  Capricorni, 

-  3 

10 

20 

52  45 

50.21 

2  33  46S. 

+.  0.25 

Fomalhaut,  - 

-  1 

11 

1 

11  12 

50.72 

21  6  27S. 

4-  0.01 

10 


TABLE  IX, 


Surfs  Epochs . 


Years. 

M.  Long. 

Long.  Perig. 

n. 

III. 

IV. 

1821 

9*  8°  48' 19" 

9*  7°  50' 43" 

920 

782 

260 

036 

1822 

9  8  34  0 

9  7  51  45 

280 

697 

886 

090 

1823 

9  8  19  40 

9  7  52  47 

640 

612 

511 

143 

1824  B. 

9  9  4  29 

9  7  53  49 

034 

530 

138 

197 

1825 

9  8  50  9 

9  7  54  51 

394 

445 

763 

251 

1826 

9  8  35  49 

9  7  55  52 

754 

360 

388 

304 

1827 

9  8  21  30 

9  7  56  54 

114 

275 

013 

358 

1828  B. 

9  9  6  18 

9  7  57  56 

508 

192 

640 

412 

j  1829 

9  8  51  59 

9  7  58  58 

868 

107 

265 

466 

!  1830 

9  8  37  39 

9  8  0  0 

228 

022 

890 

519 

|  1831 

9  8  23  19 

9  8  12 

588 

937 

515 

573 

j  1832 B. 

9  9  8  8 

9  8  2  4 

982 

855 

142 

627 

1833 

9  8  53  49 

9  8  3  6 

342 

770 

767 

681 

1834 

9  8  39  29 

9  8  4  8 

702 

684 

392 

734 

1835 

1  . 

9  8  25  9 

9  8  5  10 

062 

600 

017 

788 

1836  B. 

9  9  9  58 

9  8  6  12 

456 

517 

644 

842 

1837 

9  8  55  38 

9  8  7  13 

816 

432 

269 

895 

1838 

9  8  41  19 

9  8  8  15 

176 

347 

894 

949 

1839 

9  8  26  59 

9  8  9  17 

536 

262 

519 

003 

1840  B. 

9  9  11  48 

9  8  10  19 

930 

180 

146  1 

056 

1841 

9  8  57  28 

9  8  11  21 

290 

095 

771 

110 

1842 

9  8  43  9 

9  8  12  23 

650 

009 

397 

164 

1843 

9  8  28  49 

9  8  13  25 

010 

925 

021 

218 

1844  B. 

9  9  13  38 

9  8  14  27 

404 

843 

648 

272 

1845 

9  8  59  18 

9  8  15  29 

764 

757 

273 

325 

1846 

9  8  44  58 

9  8  16  31 

124 

673 

897 

379 

1847 

9  8  30  39 

9  8  17  33 

484 

588 

623 

433 

1848  B. 

9  9  15  27 

9  8  18  35 

878 

505 

151 

487 

1849 

9  9  18 

9  8  19  37 

238 

420 

775 

540 

1850 

9  8  46  48 

9  8  20  38 

598 

336 

400 

594 

1851 

9  8  32  28 

9  8  21  40 

958 

250 

025 

648 

1852  B. 

9  9  17  17 

9  8  22  42 

353 

168 

653 

701 

1853 

9  9  2  58 

9  8  23  44 

713 

083 

277 

755 

1874 

9  8  48  38 

9  8  24  46 

073 

998 

902 

809 

1855 

9  8  34  18 

9  8  25  48 

433 

913 

527 

863 

1856  B. 

9  9  19  7 

9  8  26  50 

827 

832 

153 

916 

1857 

9  9  4  47 

9  8  27  52 

187 

746 

779 

670 

1858 

9  8  50  28 

9  8  28  54 

547 

661 

404 

024 

1859 

9  8  36  8 

9  8  29  56 

907 

576 

029 

078 

t  1860  B. 

9  9  20  57 

9  8  30  57 

301 

494 

656 

131  J 

TABLE  X 


Sun's  Motions  for  Months. 


Months. 

Longitude. 

Per. 

I. 

11. 

111. 

N.  ; 

0* 

0°  O' 

0' 

0" 

0 

0 

0 

0 

11 

29 

0 

52 

0 

966 

997 

998 

0 

1 

0 

33 

18 

5 

47 

78 

53 

4 

0 

29 

34 

10 

5 

13 

75 

51 

4 

March, 

1 

28 

9 

11 

10 

293 

148 

101 

9 

April,  -  - 

2 

28 

42 

30 

15 

42 

226 

154 

13 

May,  -  - 

3 

28 

16 

40 

20 

59 

301 

206 

18 

June,  -  - 

4 

28 

49 

58 

26 

110 

379 

259 

22 

July,  -  - 

5 

28 

24 

8 

31 

129 

454 

510 

27 

August,  -  - 

6 

28 

57 

26 

36 

182 

531 

563 

31 

September, 

7 

29 

30 

44 

41 

233 

609 

416 

36 

October,  - 

8 

29 

4 

54 

46 

250 

684 

468 

40 

November, 

9 

29 

38 

12 

52 

300 

762 

521 

45 

December, 

10 

29 

12 

22 

57 

313 

837 

572 

49 

TABLE  XI. 

Sun's  Horary  Motion, 
Argument.  Sun’s  Mean  Anomaly. 


0* 

1* 

II* 

III* 

IV* 

V* 

0° 

2' 33" 

2'  32" 

2' 30" 

2'  28" 

2'  25" 

2' 24" 

30° 

10 

2  33 

2  32 

2  29 

2  27 

2  25 

2  23 

20 

20 

2  33 

2  31 

2  29 

2  26 

2  24 

2  23 

10 

30 

2  32 

2  30 

2  28 

2  25 

2  24 

2  23 

0 

XI* 

X* 

IX* 

VIII* 

vn* 

VI* 

TABLE  XII. 

Sun's  Semidiameter. 
Argument.  Sun’s  Mean  Anomaly. 


0* 

1* 

11* 

Ill* 

IV* 

V* 

0° 

16'  18" 

16'  15" 

16' 9" 

16'  1" 

15'  53" 

15' 48" 

30° 

10 

16  18 

16  14 

16  7 

15  58 

15  51 

15  46 

20 

20 

16  17 

16  12 

16  4 

15  56 

15  49 

15  46 

10 

30 

16  15 

16  9 

16  1 

15  53 

15  48 

15  45 

0 

XI* 

X* 

IX* 

vm* 

VII* 

VI* 

TABLE  XIII 


Sun’s  Motions  for  Days  and  Hours. 


Days 

Longitude. 

Her. 

L 

II. 

III. 

N. 

Hours. 

Long. 

1.  < 

1 

0°  0'  0" 

0" 

0 

0 

0 

0 

1 

2'  28" 

i! 

2 

0  59  8 

0 

34 

3 

2 

0 

2 

4  56 

3 

3 

1  58  17 

0 

68 

5 

3 

0 

3 

7  23 

4 

4 

2  57  25 

0 

101 

8 

5 

0 

4 

9  51 

6 

5 

3  56  33 

1 

135 

10 

7 

1 

.5 

12  19 

7 

6 

4  55  42 

1 

169 

13 

9 

1 

6 

14  47 

8 

7 

5  54  50 

1 

203 

15 

10 

1 

7 

17  15 

10 

8 

6  53  58 

1 

236 

18 

12 

1 

3 

19  43 

11 

9 

7  53  7 

1 

270 

20 

14 

1 

9 

22  11 

13 

10 

8  52  15 

1 

304 

23 

15 

1 

10 

24  38 

14 

11 

9  51  23 

2 

338 

25 

17 

1 

11 

27  6 

16 

12 

10  50  32 

2 

371 

28 

19 

2 

12 

29  34 

17 

13 

11  49  40 

2 

405 

30 

21 

2 

13 

32  2 

18 

14 

12  48  48 

2 

439 

33 

22 

2 

14 

34  30 

20 

15 

13  47  57 

2 

473 

35 

24 

2 

15 

36  58 

21 

16 

14  47  5 

3 

506 

38 

26 

2 

16 

39  26 

23 

17 

15  46  13 

3 

540 

40 

27 

2 

17 

41  53 

24 

18 

16  45  22 

3 

574 

43 

29 

2 

18 

44  21 

25 

19 

17  44  30 

3 

608 

45 

31, 

3 

19 

46  49 

27 

20 

18  43  38 

3 

641 

48 

33 

3 

20 

49  17 

28 

21 

19  42  47 

3 

675 

50 

34 

3 

21 

51  45 

30 

22 

20  41  55 

4 

709 

53 

36 

3 

22 

54  13 

31 

23 

21  41  3 

4 

743 

55 

38 

3 

23 

56  40 

32 

24 

22  40  12 

4 

777 

58 

39 

3 

24  , 

59  8 

34 

25 

23  39  20 

4 

810 

60 

41 

4 

26 

24  38  28 

4 

844 

63 

43 

4 

27 

25  37  37 

4 

878 

65 

45 

4 

28 

26  36  45 

5 

912 

68 

46 

4 

29 

27  35  53 

5 

945 

70 

48 

4 

30 

28  35  2 

5 

979 

73 

50 

4 

31 

29  34  10 

5 

13 

75 

51 

4 

TABLE  XrV. 


Sun's  Motions  for  Minutes  and  Seconds. 


.  Min. 

Long.  'Min. 

Long. 

Sec.j  Long. 

See. [Long. 

i 

i 

0'  2" 

31 

1'  16" 

1 

0" 

31 

1" 

2 

0  5 

32 

1  19 

2 

/0 

32 

1 

3 

0  7 

33 

1  21 

3 

0 

33 

1 

4 

0  10 

34 

1  24 

4 

0 

34 

1 

5 

0  12 

35 

1  26 

5 

0 

35 

1 

6 

0  15 

36 

1  29 

6 

0 

36 

1 

1  7 

0  17 

37 

1  31 

7 

0 

37 

2 

]  8 

0  20 

38 

1  34  ■ 

8 

0 

38 

2 

9 

0  22 

39 

1  36 

9 

0 

39 

2 

io- 

0  25 

40 

1  39 

10 

0 

40 

2 

11 

0  27 

41 

1  41 

11 

0 

41 

2 

12 

0  30 

42 

1  43 

12 

0 

42 

2 

13 

0  32 

43 

1  46 

13 

1 

43 

2 

14 

0  34 

44 

1  48 

14 

1 

44 

2 

15 

0  37 

45 

1  51 

15 

1 

45 

2 

16 

0  39 

46 

1  53 

16 

1 

46 

2 

i  17 

0  42 

47 

1  56 

17 

1 

47 

2 

|  18 

0  44 

48 

1  58 

18 

1 

48 

2 

i  19 

0  47 

49 

2  1 

19 

1 

49 

2 

|  20 

0  49 

50 

2  3  ' 

20 

1 

50 

2 

j  21 

0  52 

51 

2  6 

21 

1 

51 

2 

1  22- 

0  54 

52 

2  8 

22 

1 

52 

2 

23 

0  57 

53 

2  11 

23 

1 

53 

2 

24 

0  59 

54 

2  13 

24 

1 

54 

2 

25 

« 

1  2 

55 

2  16 

25 

1 

55 

2 

!  26 

1  4 

56 

2  18 

26 

1 

56 

2 

27 

1  7 

57 

2  20 

27 

1 

57 

2 

28 

1  9 

58 

2  23 

28 

1 

58 

2 

29 

1  11 

|  59 

2  25 

29 

1 

59 

2 

30 

1  14  | 

1  60 

2  28 

30 

- 4 

1 

60 

13 


14 


TABLE  XV 


Equatim  of  lltc  Sun's  Centre. 
Argument.  Sun’s  Mean  Anomaly. 


0' 

I‘ 

U‘ 

IIIs 

IV* 

v‘  '  , 

0° 

1°  59'  30" 

2°  58'  19" 

3°  40'  35" 

3°  .54'  59 ' 

3°  38'  29" 

2°  56'  13" 

1 

2  1  34 

3  0  5 

3  41  33 

3  54  55 

3  37  26 

2  54  29 

2 

2  3  37 

3  1  49 

3  42  29 

3  54  50 

3  36  22 

2  52  44 

3 

2  5  40 

3  3  32 

3  43  23 

3  54  42 

3  35  16 

!  2  50  58 

4 

2  7  44 

3  5  14 

3  44  16 

3  54  32 

3  34  8 

2  49  12 

5 

2  9  47 

3  6  54 

3  45  6 

3  54  20 

3  32  59 

2  47  24 

6 

2  11  50 

3  8  34 

3  45  55 

3  54  6 

3  31  48 

2  45  36 

7 

2  13  53 

3  10  12 

3  46  41 

3  53  50 

3  30  35 

2  43  46  . 

8 

2  15  54 

3  11  48 

3  47  26 

3  53  32 

3  29  21 

2  41  56 

9 

2  17  57 

3  13  23 

3  48  8 

3  53  11 

3  28  5 

2  40  6 

10 

2  19  59 

3  14  57 

3  48  48 

3  52  49 

3  26  48 

2  38  14 

11 

2  22  0 

3  16  30 

3  49  27 

3  52  25 

3  25  29 

2  36  22 

12 

2  24  1 

3  18  1 

3  50  3 

3  51  58 

3  24  8 

2  34  30 

13 

2  26  2 

3  19  30 

3  50  37 

3  51  30 

3  22  46 

2  32  36 

14 

2  28  2 

3  20  58 

3  51  9 

3  50  59 

3  21  23 

2  30  42 

15 

2  30  1 

3  22  24 

3  51  40 

3  50  27 

3  19  58 

2  28  48 

16 

2  32  0 

3  23  49 

3  52  8 

3  49  52 

3  18  32 

2  26  53 

17 

2  33  58 

3  25  12 

3  52  34 

3  49  16 

3  17  4 

2  24  58 

18 

2  35  55 

3  26  33 

3  52  57 

3  48  37 

3  15  36 

2  23  2 

19 

2  37  52 

3  27  53 

3  53  19 

3  47  57 

3  14  6 

2  21  6 

20 

2  39  48 

3  29  11 

3  53  39 

3  47  15 

3  12  34 

2  19  9 

21 

2  41  43 

3  30  27 

3  53  56 

3  46  31 

3  11  1 

2  17  12 

22 

2  43  38 

3  31  42 

3  54  12 

3  45  45 

3  '9  27 

2  15  15 

23 

'  2  45  31 

3  32  55 

3  54  25 

3  44  56 

3  7  52 

2  13  17 

24 

2  47  24 

3  34  6 

3  54  36 

3  44  7 

3  6  15 

2  11  20 

25 

2  49  16 

3  35  15 

3  54  45 

3  43  15 

3  4  38 

2  9  22 

26 

2  51  6 

3  36  23 

3  54  52- 

3  42  21 

3  2  59 

2  7  24 

27 

2  52  56 

3  37  29 

3  54  57 

3  41  26 

3  1  19 

2  5  25 

28 

2  54  45 

3  38  33 

3  55  0 

3  40  28 

2  59  38 

2  3  27 

29 

2  56  33 

3  39  35 

3  55  0 

3  39  29 

2  57  56 

2  1  28 

30 

2  58  19 

3  40  35 

3  54  59 

3  38  29 

2  56  13 

1  59  30 

4. 


TABLE  XV, 


15 


Equation  of  the  Sun's  Centre . 
Argument.  Sun*s  Mean  Anomaly. 


VI* 

VII* 

VIII* 

IX* 

X* 

XI* 

0° 

1°  59'  30" 

1°  2'  47" 

0°  20' 31" 

0° 

1  4' 

1" 

0°  18'  25" 

1°  0'  41* 

1 

1 

57 

32 

1 

1 

4 

0 

19 

31 

0 

4 

0 

0 

19 

25 

1 

2 

27 

2 

1 

55 

33 

0 

59 

22 

0 

18 

32 

0 

4 

0 

0 

20 

27 

1 

4 

15 

3 

1 

53 

35 

0 

57 

41 

0 

17 

34 

0 

4 

3 

0 

21 

31 

1 

6 

4 

4 

1 

51 

36 

0 

56 

1 

0 

16  39 

0 

4 

8 

0 

22  37 

1 

7 

54 

5 

1 

49 

38 

0 

54 

22 

0 

15 

45 

0 

4 

15 

0 

23 

45 

1 

9 

44 

6 

1 

47 

40 

0 

52 

45 

0 

14 

53 

0 

4 

24 

0 

24 

54 

1 

11 

36 

7 

1 

45 

43 

0 

51 

8 

0 

14 

4 

0 

4 

35 

0 

26 

5 

1 

13 

29 

8 

1 

43 

45 

0 

49 

33 

0 

13 

15 

0 

4 

48 

0 

27 

18 

1 

15 

22 

9 

1 

41 

48 

0 

47 

59 

0 

12 

29 

0 

5 

4 

0 

28 

33 

1 

17 

17 

10 

1 

39 

51 

0 

46 

26 

0 

11 

45 

0 

5 

21 

0 

29  49 

1 

19 

12 

11 

1 

37 

54 

0 

44 

54 

0 

11 

3 

0 

5 

41 

0 

31 

7 

1 

21 

8 

12 

1 

35 

58 

0 

43 

24 

0 

10 

23 

0 

6 

3 

0 

32 

27 

1 

23 

5 

13 

1 

34 

2 

0 

41 

56 

0 

9 

44 

0 

6 

26 

0 

33 

48 

1 

25 

2 

14 

1 

32 

7 

0 

40 

28 

0 

9 

8 

0 

6 

52 

0 

35 

11 

1 

27 

0 

15 

1 

30 

12 

0 

39 

2 

0 

8 

33 

0 

7 

20 

0 

36 

36 

1 

28 

59 

16 

1 

28 

18 

0 

37  37 

0 

8 

1 

0 

7 

51 

0 

38 

2 

1 

30 

58 

17 

1 

26 

24 

0 

36 

14 

0 

7 

30 

0 

8 

23 

0  39  30 

1 

32 

58 

18 

1 

24 

30 

0 

34  52 

0 

7 

2 

0 

8 

57 

0 

40 

59 

1 

34 

59 

19 

1 

22 

38 

0 

33 

31 

0 

6 

35 

0 

9 

33 

0 

42 

30 

1 

37 

0 

20 

1 

20 

46 

0  32 

12 

0 

6 

11 

0 

10 

12 

0 

44 

3 

1 

39 

1 

21 

1 

18 

54 

0 

30 

55 

0 

5 

49 

0 

10 

52 

0 

45 

37 

1 

41 

3 

22 

1 

17 

4 

0 

29 

39 

0 

5 

28 

0 

11 

34 

0 

47 

12 

1 

43 

6 

23 

1 

15 

14 

0 

28 

25 

0 

5 

10 

0 

12 

19 

0 

48 

48 

1 

45 

7 

24 

1 

13 

24 

0 

27 

12 

0 

4 

54 

0 

13 

5 

0 

50 

26 

1 

47 

10 

25 

1 

11 

36 

0 

26 

1 

0 

4  40 

0 

13 

54 

0 

52 

6 

1 

49 

13 

26 

1 

9  48 

0 

24 

52 

0 

4 

28 

0 

14 

44 

0 

53 

46 

1 

51 

16 

27 

1 

8 

2 

0 

23 

44 

0 

4 

18 

0 

15 

37 

0 

55 

28 

1 

53 

20 

28 

1 

6 

16 

0 

22 

38 

0 

4 

10 

0 

16 

31 

0 

57 

11 

1 

55 

23 

29 

1 

4 

31 

0 

21 

34 

0 

4 

5 

0 

17  27 

0 

58 

55 

1 

57 

26 

30 

1 

2 

47 

0  20 

31 

0 

4 

1 

0 

18  25 

1 

0 

41 

1 

59 

30 

16 


TABLE  KYI- 


TABLE  XVII 


Small  Equations  of  Sun's  Longitude. 


j  Arg. 

I- 

u. 

III. 

5  Al-g. 

I. 

U. 

111. 

0 

10" 

10" 

10"| 

500 

10" 

10" 

10" 

30 

10 

11 

9 

510 

10 

10 

9 

(  20 

11 

11 

9 

520 

9 

10 

8 

30 

11 

12 

8 

530 

i  9 

10 

7 

40 

11 

13 

8 

540 

9 

10 

7 

j  50 

12 

14 

7 

550 

8 

10 

6 

j  . 

|  60 

12 

14 

7 

560 

8 

9 

5 

70 

12 

15 

7 

570 

8 

9 

4 

80 

13 

15 

7 

580 

7 

9 

3 

90 

13 

16 

7 

590 

7 

9 

3 

100 

13 

16 

7 

600 

7 

9 

2 

110 

14 

17 

7 

610 

6 

8 

1 

120 

14 

17 

7 

620 

6 

8 

1 

130 

14 

18 

8 

630 

6 

8 

1 

140 

15 

18 

8 

640 

5 

7 

0 

150 

15 

18 

9 

650 

5 

7 

0 

160 

15 

18 

9 

660 

5 

6 

0 

170 

15 

18 

10 

670 

5 

6 

1 

180 

15 

18 

10 

680 

5 

6 

1 

190 

16 

18 

11 

690 

4 

5 

2 

200 

16 

18 

11 

700 

4 

5 

2 

210 

16 

18 

12 

710' 

4 

4 

o 

O 

J  220 

16 

18 

12 

720 

4 

4 

3 

j  230 

16 

18 

13 

730 

4 

4 

4 

!  240 

16 

17 

14 

740 

4 

3 

5 

|  250 

16 

17 

14 

750 

4 

3 

6 

!  260 

16 

17 

15 

760 

4 

3 

6 

!  270 

16 

16 

16 

770 

4 

2 

7 

;  280 

16 

16 

17 

780 

4 

2 

8 

j  290 

16 

16 

17 

790 

4 

o 

8 

;  300 

16 

15 

18 

800 

4 

2 

9 

i  310 

16 

15 

18 

810 

4 

2 

9 

j  520 

15 

14 

19 

820 

5 

2 

10 

!  330 

35 

14 

19 

830 

5 

2 

10 

j  340 

15 

14 

20 

840 

5 

2 

11 

j  350 

15 

13 

20 

850 

5 

2 

11 

360 

15 

13 

20  ‘ 

860 

5 

2 

12 

370 

14 

12 

19 

870 

6 

2 

12 

380 

14 

12 

19 

880 

6 

3 

13 

390 

14 

12 

19 

890 

6 

3 

13 

(  400 

13 

11 

18 

900 

7 

4 

13 

410 

13 

11 

17 

910 

7 

4 

13 

;  420 

13 

11 

17 

920 

7 

5 

13 

{  430 

12 

11 

16 

930 

8 

5 

13 

;  440 

12 

11 

15 

940 

8 

6 

13 

j  450 

12 

10 

14 

950 

8 

6 

13 

j  460 

11 

10 

13 

960 

9 

7 

12 

!  470 

11 

10 

13 

970 

9 

8 

12 

:  480 

11 

10 

12 

980 

9 

9 

11 

'  490 

10 

10 

11 

990 

10 

9 

11 

j  500 

10 

10 

10 

1000 

10 

10 

10 

Mean  Obliquity  of  the 
Ecliptic. 


Years. 

M.Obliqu. 

1821 

23°  27'  46" 

1822 

23  27  46 

1823 

23  27  45 

1824 

23  27  44 

1825 

23  27  44 

1826 

23  27  43 

1827 

23  27  43 

1828 

23  27  42 

1829 

23  27  42 

1830 

23  27  41 

1831 

23  27  41 

1832 

23  27  40 

1833 

23  27  40 

1834 

23  27  39 

1835 

23  27  39 

1836 

23  27  38 

1837 

23  27  38 

1838 

23  27  37 

1839 

23  27  37 

1840 

23  27  36 

1841 

23  27  36  | 

1842 

23  27  35 

1843 

23  27  35 

1844 

23  27  34 

1845 

23  27  34 

1846 

23  27  33 

1847 

23  27  33 

1848 

23  27  32 

1849 

23  27  31 

1850 

23  27  30 

1851 

23  27  30 

1852 

23  27  29 

1853 

23  27  29 

1854 

23  27  28 

1855 

23  27  28 

1856 

23  27  27 

1857 

23  27  27 

1858 

23  27  26  . 

1859 

23  27  26 

1860 

23  27  25 

TABLE  XVIIL 


17 

Nutation. 


Argument.  Supplement  of  the  Node,  or  N. 


18 


TABLE  XIX 


Equation  oj  Time ,  to  convert  Apparent  into  Mean. 
Argument.  Sun’s  Mean  Longitude. 


0* 

1* 

11* 

III* 

IV* 

V* 

j 

o 

m.  sec. 

m.  sec. 

m.  sec. 

m.  sec. 

m.  sec. 

m.  sec.! 

0 

4-  6  21 

—  1  55 

—  3  32 

4-  1  50 

+  66 

+  2  17  i 

1 

6  2 

2  6 

3  26 

2  3 

6  7 

2  1 

2 

5  43 

2  17 

3  20 

2  16 

6  7 

1  45  1 

3 

5  25 

2  27 

3  13 

2  29 

6  7 

1  28  l 

4 

5  6 

2  37 

3  6 

2  42 

6  6 

1  11  i 

5 

4  47 

2  46 

2  59 

2  54 

6  5 

0  53 

6 

4  28 

2  55 

2  51 

3  6 

6  3 

0  35 

7 

4  9 

3  3 

2  42 

3  18 

6  0 

+  0  17 

8 

3  51 

3  11 

2  33 

3  30 

5  57 

—  02 

9 

3  33 

3  18 

2  24 

3  41 

5  53 

0  21 

10 

3  14 

3  24 

2  14 

3  52 

5  49 

0  40 

11 

2  56 

3  30 

2  4 

4  3 

5  44 

1  0 

12 

2  38 

3  36 

1  54 

4  14 

5  38 

1  19 

13 

2  20 

3  40 

1  43 

4  24 

5  32 

1  39 

14 

2  2 

3  45 

1  32 

4  33 

5  25 

2  0 

15 

1  45 

3  48 

1  21 

4  43 

5  17 

2  20 

16 

1  28 

3  51 

1  9 

4  52 

5  9 

2  41 

17 

1  11 

3  53 

0  57 

5  0 

5  0 

3  2 

18 

0  55 

3  55 

0  45 

5  8 

4  51 

3  23 

19 

0  38 

3  56 

0  33 

5  16 

4  41 

3  44 

20 

0  22 

3  57 

0  20 

5  23 

4  31 

4  5 

21 

4-0  7 

3  57 

—  07 

5  30 

4  20 

4  26 

22 

—  08 

3  57 

-f  0  5 

5  36 

4  8 

4  47  . 

23 

0  23 

3  56 

0  18 

5  42 

3  56 

5  9 

24 

0  37 

3  54 

0  32 

5  47 

3  43 

5  30 

25 

0  51 

3  51 

0  45 

5  52 

3  30 

5  51 

26 

1  4 

3  49 

0  58 

5  56 

3  17 

6  13 

.27 

1  17 

3  45 

1  11 

5  59 

3  3 

6  34 

28- 

1  30 

3  41 

1  24 

6  2 

2  48 

6  56 

29 

1  43 

3  37 

1  37 

6  4 

2  33 

7  17 

30 

—  1  55 

—  3  32 

4-  1  50 

+  66 

+  2  17 

—  7  38 

TABLE  XIX 


19 


Equation  of  Time ,  to  convert  A pparent  into  Mean. 
Argument.  Sun’s  Mean  Longitude. 


VI* 

Vii* 

VIIIs 

IXs 

Xs 

Xi‘ 

0 

m.  sec. 

m.  sec. 

m.  sec. 

m.  sec. 

m.  sec. 

m.  sec. 

0 

—  7  38 

—  15  35 

—  13  24 

—  0  27 

4  12  7 

-f  13  50 

1 

7  59 

15  43 

13  7 

+  04 

12  22 

13  42 

2 

8  19 

15  49 

12  48 

0  34 

12  36 

13  33 

3 

8  40 

15  56 

12  29 

1  4 

12  50 

13  23 

4 

9  1 

16  1 

12  10 

1  35 

13  3 

13  13 

5 

9  21 

16  5 

11  49 

2  4 

13  15 

13  2 

6 

9  41 

16  9 

11  28 

2  34 

13  27 

12  51 

7 

10  0 

16  12 

11  6 

3  4 

13  37 

12  39 

8 

10  20 

16  14 

10  43 

3  33 

13  47 

12  26 

9 

10  39 

16  15 

10  20 

4  2 

13  55 

12  13 

10 

10  58 

16  16 

9  56 

4  30 

14  3 

12  0 

11 

11  16 

16  15 

9  31 

4  59 

14  10 

11  46 

12 

11  34 

16  14 

9  6 

5  26 

14  16 

11  32 

13 

11  52 

16  12 

8  40 

5  54 

14  22 

11  17 

14 

12  9 

16  9 

8  14 

6  21 

14  26 

11  2 

15 

12  26 

16  5 

7  47 

6  47 

14  30 

10  46 

16 

12  43 

16  1 

7  20 

7  13 

14  33 

10  30 

17 

12  59 

15  55 

6  52 

7  38 

14  35 

10  14 

18 

13  14 

15  49 

6  24 

8  3 

14  36 

9  58 

19 

13  29 

15  42 

5  56 

8  27 

14  36 

9  41 

20 

13  44 

15  33 

5  27 

8  50 

14  35 

9  23 

21 

13  58 

15  24 

4  58 

9  13 

14  34 

9  6 

22 

14  11 

15  14 

4  28 

9  35 

14  32 

8  48 

23 

14  24 

15  3 

3  59 

9  57 

14  30 

8  31 

24 

14  36 

14  52 

3  29 

10  18 

14  26 

8  13 

25 

14  48 

14  39 

2  59 

10  38 

14  22 

7  54 

26 

14  58 

14  26 

2  28 

10  57 

14  17 

7  36 

27 

15  9 

14  12 

1  58 

11  16 

14  11 

7  17 

28 

15  18 

13  57 

1  28 

11  33 

14  5 

6  59 

29 

15  27 

13  41 

0  57 

11  50 

13  58 

6  40 

30 

—  15  35 

—  13  24 

—  0  27 

4  12  7 

4-  13  50 

+  6  21 

/ 


20- 


S  ABLE  XX. 


Moon's  Epochs, 


Years.  1 

1  1 

2 

3 

4 

5 

6 

7 

8 

9 

1821 

0027 

8365 

5389 

1368 

6970 

7714 

6319 

7024 

7800 

1822 

0020 

5573 

5054 

6112 

9441 

3512 

7380 

9481 

6664 

1823 

0012 

2782 

4720 

0856 

1913 

9309 

8440 

1958 

5528 

1821B. 

0033 

0640 

5426 

5887 

4720 

5478 

9559 

4787 

4417 

1825 

0026 

7849 

5092 

0631 

7192 

1276 

0619 

7243 

3280 

1826 

0018 

5057 

4758 

5375 

9663 

7073 

1680 

9701 

2144 

1827 

0011 

2265 

4424 

0119 

2135 

2871 

2740 

2158 

1008 

1828  B. 

0032 

0124 

5129 

5150 

4942 

9040 

3859 

5007 

9896 

1829 

0024 

7532 

4795 

9894 

7414 

4837 

4919 

7463 

8760 

1830 

0017 

4541 

4461 

4638 

9885 

0635 

5979 

9921 

7623 

1831 

0010 

1749 

4127 

9381 

2357 

6432 

7040 

2378 

6487 

1832  B. 

0030 

9607 

4833 

4412 

5164 

2601 

8158 

5226 

5376 

1833 

0023 

6816 

4499 

9156 

7636 

8399 

9219 

7683 

4239 

1834 

0016 

4024 

4164 

3900 

0107 

4196 

0279 

0140 

3103 

1835 

0009 

1232 

3830 

8644 

2579 

9993 

1340 

2598 

1967 

1836  B. 

0029 

9091 

4536 

3675 

5386 

6163 

2458 

5446 

0856 

1837 

0022 

6299 

4202 

8419 

7858 

1960 

3518 

7903 

9719 

1838 

0015 

3508 

3868 

3163 

0329 

7757 

4579 

0360 

8583 

1839 

0008 

0716 

3534 

7907 

2801 

3555 

5639 

2818 

7447 

1840  B. 

0028 

8575 

4239 

2938 

5608 

9724 

6758 

5666 

6335 

1841 

0021 

5783 

3906 

7682 

8080 

5522 

7818 

8123 

5199 

1842 

0014 

2991 

3571 

2425 

0551 

1319 

8879 

0580 

4062 

1843 

0007 

0200 

3237 

7169 

3023 

7116 

9939 

3038 

2926 

1844  B. 

0027 

8058 

3943 

2200 

5830 

3286 

1058 

5886 

1815* 

1845 

0020 

5266 

3609 

6944 

8302 

9083 

2118 

8343 

0678 

1846 

0013 

2475 

3275 

1688 

0773 

4880 

3179 

0800 

9542 

1847 

0006 

9683 

2941 

6432 

3245 

0678 

4239 

3257 

8406 

1848  B. 

0026 

7542 

3646 

1463 

6052 

6847 

5358 

6106 

7295 

1849 

0019 

4750 

3312 

6207 

8524 

2644 

6418 

8563 

6158 

1850 

0012 

1958 

2978 

0951 

0995 

8442 

7479 

1020 

5022 

1851 

0005 

9167 

2644 

5695 

3467 

4239 

8539 

3477 

3885 

1852  B. 

0025 

7025 

3350 

0726 

6274 

0408 

9658 

6326 

2774 

1853 

0018 

4233 

3016 

5469 

8746 

6206 

0718 

8782 

1637 

1854 

0011 

1442 

2681 

0213 

1217 

2003 

1778 

1240 

0501 

(  1855 

0004 

8650 

2347 

4957 

3689 

7801 

2839 

3697 

9365 

1856  B. 

0024 

6509 

3053 

9988 

6496 

3970 

3957 

6546 

8254 

1857 

0017 

3717 

2719 

4732 

8968 

9767 

5018 

9002 

7117 

1858 

0010 

0925 

2385 

9476 

1439 

5565 

6078 

1460 

5981 

1859 

0003 

8134 

2051 

4220 

3911 

1362 

7139 

3917 

4845 

’  1860  B. 

0023 

5992 

2756 

9251 

6718 

7531 

8257 

6765 

3734 

TABLE  XX 


21 


Maori's  Epochs . 


Years. 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

2o 

1821 

620 

917 

842 

142 

979 

067 

923 

331 

134 

036 

036 

1822 

226 

278 

562 

615 

172 

208 

282 

684 

609 

090 

202 

1823 

833 

639 

281 

088 

366 

348 

641 

036 

084 

143 

369 

1824  B. 

509 

030 

070 

595 

659 

519 

037 

431 

585 

197 

537 

1825 

116 

391 

790 

068 

853 

659 

397 

783 

060 

251 

703 

1826 

722 

752 

510 

541 

047 

800 

7o  6 

136 

536 

304 

869 

1827 

329 

113 

229 

014 

241 

940 

115 

488 

Oil 

358 

036 

1828  B. 

005 

505 

019 

521 

533 

111 

511 

883 

512 

412 

204 

1829 

612 

866 

738 

994 

727 

251 

871 

235 

987 

466 

370 

1830 

219 

226 

458 

468 

921 

392 

230 

588 

462 

519 

536 

1831 

825 

587 

177 

940 

115 

532 

589 

940 

937 

573 

703 

1832  B. 

502 

979 

967 

447 

408 

704 

985 

335 

438 

627 

871 

1833 

108 

340 

687 

920 

602 

844 

345 

688 

913 

681 

037 

1834 

715 

701 

406 

393 

796 

984 

704 

040 

388 

734 

203 

1835 

321 

061 

125 

866 

989 

124 

063 

393 

863 

788 

370 

1836  B. 

998 

453 

915 

373 

282 

296 

459 

787 

364 

842 

538 

1837 

605 

814 

635 

846 

476 

436 

819 

140 

840 

895 

704 

1838 

211 

175 

354 

319 

670 

576 

178 

492 

315 

949 

870 

1839. 

818 

536 

074 

792 

864 

716 

537 

845 

790 

003 

037 

1840  B. 

494 

927 

863 

299 

157 

888 

933 

239 

291 

05fe 

205 

1841 

101 

288 

583 

772 

351 

028 

293 

592 

766 

110 

371 

1842 

707 

649 

302 

245 

544 

168 

652 

944 

241 

164 

537 

1843 

314 

010 

022 

718 

738 

308 

012 

297 

716 

218 

704 

1844  B. 

990 

402 

811 

225 

031 

480 

407 

691 

217 

272 

872 

1845 

597 

763 

531 

698 

225 

620 

767 

044 

692 

325 

038 

1846 

203 

123 

250 

171 

419 

760 

126 

396 

167 

379 

204 

1847 

810 

484 

970 

644 

613 

901 

486 

749 

643 

433 

371 

1848  B. 

486 

876 

759 

151 

905 

072 

881 

143 

144 

487 

539 

1849 

093 

237 

479 

624 

099 

212 

241 

496 

619 

540 

705 

1850 

700 

597 

199 

097 

293 

352 

600 

848 

094 

594 

871 

1851 

306 

958 

918 

570 

487 

493 

960 

201 

569 

648 

038 

1852  B. 

983 

350 

707 

077 

780 

664 

355 

595 

070 

701 

206 

1853 

589 

711 

427 

550 

974 

804 

715 

948 

545 

755 

372 

1854 

196 

072 

147 

023 

168 

944 

074 

300 

020 

809 

539 

1855 

802 

432 

866 

496 

361 

085 

434 

653 

495 

863 

705 

1856  B 

479 

824 

656 

003 

654 

256 

829 

047 

996 

916 

873 

1857 

086 

185 

375 

476 

848 

396 

189 

400 

471 

970 

039 

1858 

692 

546 

095 

949 

042 

537 

548 

752 

947 

024 

206 

1859 

299 

907 

814 

422 

236 

677 

908 

105 

422 

078 

372 

1860  B. 

975 

298 

604 

929 

529 

848 

303 

499 

923 

131 

540 

TABLE  XX 


Moon's  Epochs . 


Years. 

Evection. 

Anomaly. 

Variation. 

Longitude. 

1821 

1822 

1823 

1824  B. 

1825 

1*19*  43' 47" 
7  10  15  16 

1  0  46  45 

7  2  37  15 

0  23  8  44 

8*  9®  54'  17" 
11  8  37  37 

2  7  20  57 

5  19  8  11 

8  17  51  31 

10*22°  4'  2" 

3  1  41  27 

7  11  18  51 

0  3  7  43 

4  12  45  7 

8*  2°  7'  41 ' 
0  11  30  46 

4  20  53  51 

9  13  27  31 

1  22  50  37 

1826 

1827 

1828  B. 

1829 

1830 

6  13  40  14 

0  4  11  44 

6  6  2  13 
11  26  33  43 

5  17  5  12 

11  16  34  50 

2  15  18  10 

5  27  5  24 

8  25  48  44 
11  24  32  4 

8  22  22  32 

1  1  59  56 

5  23  48  49 

10  3  26  13 

2  13  3  38 

6  2  13  42 

10  11  36  47 

3  4  10  27 

7  13  33  32 

11  22  56  37 

1831 

1832  B. 

1833 

1834 

1835 

11  7  36  41 

5  9  27  11 
10  29  58  40 

4  20  30  11 
10  11  1  40 

2  23  16  24 

6  5  2  38 

9  3  45  58 

0  2  29  18 

3  1  12  38 

6  22  41  3 
11  14  29  54 

3  24  7  20 

8  3  44  44 

0  13  22  9 

4  2  19  42 

8  24  53  23 

1  4  16  28 

5  13  39  33 

9  23  2  38 

1836  B. 

1837 
;  1838 

1839 

1840  B. 

4  12  52  9 
10  3  23  39 

3  23  55  9 

9  14  26  38 

3  16  17  8 

6  12  59  52 

9  11  43  12 

0  10  26  32 

3  9  9  53 

6  20  57  7 

5  5  11  0 

9  14  48  26 

1  24  25  50 

6  4  3  15 
10  25  52  8 

2  15  36  19 

6  24  59  24 

11  4  22  29 

3  13  45  35 

8  6  19  15 

1841 

1842 

1843 

1844  B. 

1845 

9  6  48  37 

2  27  20  7 

8  17  51  37 

2  19  42  7 

8  10  13  36 

9  19  40  27 

0  18  23  47 

3  17  7  7 

6  28  54  22 

9  27  37  42 

3  5  29  32 

7  15  6  57 
11  24  44  22 

4  16  33  14 

8  26  10  39 

0  15  42  21 

4  25  5  26 

9  4  28  31 

1  27  2  12 

6  6  25  17 

1846 

1847 

1848  B. 

1849 

1850 

2  0  45  6 

7  21  16  35 

1  23  7  5 

7  13  38  35 

1  4  10  4 

0  26  21  2 

3  25  4  23 

7  6  51  37 
10  5  34  57 

1  4  18  18 

1  5  48  4 

5  15  25  29 
10  7  14  21 

2  16  51  46 

6  26  29  11 

10  15  48  23 

2  25  11  28 

7  17  45  8 

11  27  8  14 

4  6  31  20 

1851 

1852  B. 

1853 

1854 

1855 

6  24  41  35 

0  26  32  5 

6  17  3  34 

0  7  35  4 

5  28  6  33 

4  3  1  38 

7  14  48  53 
10  13  32  13 

1  12  15  34 

4  10  58  54 

11  6  6  36 

3  27  55  29 

8  7  32  53 

0  17  10  19 

4  26  47  43 

8  15  54  25 

1  8  28  6 

5  17  51  11 

9  27  14  17 

2  6  37  22 

1856  B. 
j  1857 
1858 
j  1859 
|  1860  B. 

11  29  57  3 

5  20  28  33 
11  11  0  2 

5  1  31  33 
11  3  22  3 

7  22  46  9 
10  21  29  29 

1  20  12  50 

4  18  56  10 

8  0  43  25 

9  18  36  36 

1  28  14  1 

6  7  51  26 
10  17  28  52 

3  9  17  44 

6  29  11  3 
11  8  34  9 

3  17  57  14 

7  27  20  20 

0  19  54  0 

TABLE  XX 


23 


Moon's  Epochs . 


Years. 

Supp.  of  Node 

11. 

V. 

i  V1 

VII. 

VIII. 

IX. 

X. 

1821 

0s  13°  3'  29" 

0*  27°  41' 

706 

711 

074 

079 

637 

596 

1822 

1  2  23  11 

4  18  13 

120 

124 

382 

386 

717 

536 

1823 

1  21  42  56 

8  8  45 

533 

536 

689 

692 

796 

475 

1824  B. 

2  11  5  47 

0  10  26 

981 

988 

026 

032 

912 

420 

1825 

3  0  25  29 

4  0  58 

395 

401 

333 

338 

992 

359 

1826 

3  19  45  11 

7  21  30 

809 

813 

641 

645 

072 

299 

1827 

4  9  4  53 

11  12  2 

223 

225 

949 

951 

151 

238 

1828  B. 

4  28  27  46 

3  13  43 

670 

677 

285 

291 

267 

182 

1829 

5  17  47  29 

7  4  15 

084 

090 

592 

597 

347 

122 

1830 

6  7  7  11 

10  24  47 

498 

502 

900 

904 

427 

062 

1831 

6  26  26  53 

2  15  19 

912 

914 

208 

210 

506 

001 

1832  B. 

7  15  49  46 

6  17  0 

360 

366 

545 

550 

622 

945  1 

1833 

8  5  9  28 

10  7  32 

774 

779 

852 

856 

702 

885  ] 

1834 

8  24  29  11 

1  28  4r 

187 

191 

159 

163 

782 

825  ", 

1835 

9  13  48  53 

5  18  36 

601 

603 

467 

469 

861 

764  j 

1836  B. 

10  3  11  46 

9  20  18 

048 

055 

804 

809 

977 

708 

1837 

10  22  31  28 

1  10  50 

463 

468 

111 

116 

057 

648  : 

1838 

11  11  51  10 

5  1  22 

876 

880 

419 

423 

137 

588 

1839 

0  1  10  52 

8  21  54 

290 

292 

726 

729 

217 

527 

1840  B. 

0  20  33  45 

0  23  35 

738 

744 

063 

069 

332 

471 

1841 

1  9  53  28 

4  14  7 

152 

157 

370 

375 

412 

411 

1842 

1  29  13  10 

8  4  39 

566 

569 

678 

682 

492 

350 

1843 

2  18  32  52 

11  25  11 

980 

980 

986 

988 

572 

290 

1844  B. 

3  7  55  45 

3  26  52 

427 

433 

322 

328 

687 

234 

1845 

3  27  15  27 

7  17  24 

840 

846 

629 

634 

767 

174 

1846 

4  16  35  9 

11  7  56 

254 

258 

937 

941 

847 

113 

1847 

5  5  54  52 

2  28  38 

668 

670 

245 

247 

927 

053 

1848  B. 

2  25  17  45 

7  0  9 

116 

122 

582 

587 

042 

997 

1849 

6  14  37  27 

10  20  41 

531 

535 

889 

893 

122 

937  j 

1850 

7  3  57  9 

2  11  13 

944 

947 

196 

200 

202 

876  j 

i 

1851 

7  23  16  51 

6  1  45 

358 

359 

504 

506 

282 

816  : 

1852  B. 

8  12  39  44 

10  3  27 

806 

811 

841 

846 

398 

760 

1853 

9  1  59  26 

1  23  59 

220 

223 

148 

152 

477 

700 

1854 

9  21  19  9 

5  14  31 

634 

636 

456 

459 

557 

639 

1855. 

10.  10  38  51 

9  5  3 

047 

048 

763 

765 

637 

579 

1856  B. 

11  0  1  44 

1  6  44 

495 

500 

100 

105 

753 

523 

1857 

11  19  21  26 

4  27  16 

909 

912 

407 

411 

832 

463 

1858 

0  8  41  8 

8  17  48 

323 

325 

715 

718 

912 

402 

1859 

0  28  0  51 

0  8  20 

736  1 

737 

023 

024 

992 

342 

1860  B. 

1  17  23  43 

4  10  1  1 

184  1 

189 

359 

364 

108 

286 

v 


TABLE  XXI 


24 


Moon's  Motions  for  Months. 


Months 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Tnn  C  Com. 

0000 

0000 

0000 

0000 

0000 

0000 

0000 

0000 

0000 

Jan-  l  His. 

9973 

9350 

8960 

9713 

9664 

9628 

9942 

9610 

9976 

,  C  Com. 

849 

146 

2246 

8896 

402 

1533 

1789 

2099 

753 

Feb-  inis. 

821 

9497 

1205 

8609 

66 

1161 

1731 

1709 

729 

March, 

1615 

8343 

1371 

6931 

9797 

1951 

3404 

3027 

1433 

April,  -  - 

2464 

8490 

3616 

5827 

199 

3484 

5193 

5126 

2186 

May,  -  - 

3285 

7986 

4822 

4436 

265 

4646 

6924 

6835 

2914 

June,  -  - 

4134 

8133 

7067 

3332 

666 

6179 

8713 

8934 

3667 

July,  -  - 

4955 

762918273 

1942 

732 

7341 

444 

643 

4396 

August,  -  - 

5804 

7776 

518 

838 

1134 

8874 

2233 

2742 

5148 

September, 

6653 

7922 

2764 

9734 

1536 

408 

4021 

4842 

5901 

October,  - 

7474 

7419 

3969 

8343 

1602 

1569 

5752 

6550 

6630 

November, 

8323 

7565 

6215 

7239 

2004 

3102 

7541 

8649 

7382 

December, 

914417062 

7420 

5848(2070 

4264 

9272 

358 

8111 

TABLE  XXI. 

Moons  Motions  for  Months. 


Months. 

Evection. 

Anomaly. 

Variation. 

VI. 

Long. 

To„  C  Com. 
Jan'  tills. 

0* 

0°  0' 

0" 

0* 

0°  O' 

0" 

0‘ 

0°  0' 

0" 

0* 

0°  0' 

0" 

11 

18 

41 

1 

11 

16 

56 

6 

11 

17 

48 

33 

11 

16 

49 

25 

,,  ,  C  Com. 
Feb  t»is. 

11 

20 

48 

42 

1 

15 

0 

53 

0 

17 

54 

48 

1 

18 

28 

6 

11 

9 

29 

43 

1 

1 

56 

59 

0 

5 

43 

21 

1 

5 

17 

31 

March, 

10 

7 

40 

26 

1 

20 

50 

4 

11 

29 

15 

15 

1 

27 

24 

27 

April,  *  - 

9 

28 

29 

8 

3 

5 

50 

57 

0 

17 

10 

3 

3 

15 

52 

32 

May,  -  -  - 

9 

7 

58 

51 

4 

7 

47 

56 

0 

22 

53 

24 

4 

21 

10 

3 

June,  -  - 

8 

28 

47 

33 

5 

22 

48 

49 

1 

10 

48 

11 

6 

9 

38 

9 

July,  -  -  - 

8 

8 

17 

16 

6 

24 

45 

48 

1 

16 

31 

32 

7 

14 

55 

40 

August, 

7 

29 

5 

59 

8 

9 

46 

42 

2 

4 

26 

20 

9 

3 

23 

46 

September, 

7 

19 

54 

41 

9 

24 

47 

35 

2 

22 

21 

7 

10 

21 

51 

52 

October,  - 

6 

29 

24 

24 

10 

26 

44 

34 

2 

28 

4 

28 

11 

27 

9 

22 

November, 

6 

20 

13 

6 

0 

11 

45 

27 

3 

15 

59 

16 

1 

15 

37 

28 

December, 

5 

29 

42 

49 

1 

13 

42 

26 

3 

21 

42 

37 

2 

20 

54 

59 

TABLE  XXL 


25 


Maori's  Motions  for  Months . 


j  Months. 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19  . 

20 

r  ( Com. 

000 

000 

000 

000 

000 

000 

000 

000 

000 

000 

000 

Jan-  Ui,. 

930 

969 

930 

966 

901 

969 

963 

958 

974 

000 

000 

,,  ,  C  Com. 

175 

965 

184 

59 

74 

946 

135 

304 

805 

5 

14 

leb-  \  Bis. 

105 

934 

114 

25 

975 

916 

98 

262 

779 

5 

14 

March,  r 

139 

836 

157 

16 

851 

801 

159 

482 

532 

9 

27 

April,  -  - 

314 

801 

342 

76 

925 

747 

294 

786 

336 

13 

41 

May,  -  - 

419 

735 

456 

101 

899 

663 

392 

47 

115 

18 

55 

June,  -  - 

593 

700 

640 

160 

973 

609 

527 

351 

920 

22 

69 

July,  -  - 

698 

634 

754 

185 

948 

525 

625 

613 

699 

27 

83 

August,  -  - 

873 

599 

938 

245 

22 

471 

759 

917 

503 

31 

97 

September, 

48 

563 

123 

304 

96 

417 

894 

221 

308 

36 

111 

October,  - 

152 

497 

237 

329 

71 

333 

992 

483 

87 

40 

125 

November, 

327 

462 

421 

388 

145 

279 

127 

787 

892 

45 

139 

December, 

432 

396 

535 

414  1120 

194 

225 

49 

670 

49 

153 

TABLE  XXI. 

Moon's  Motions  for  Months. 


Months. 

Supp. 

of  Node 

11. 

V. 

VI. 

VII. 

VIII. 

IX. 

X. 

,  C  Com. 

0» 

0°  0' 

0" 

0‘ 

0°  0' 

000 

000 

000 

000 

000 

000 

Jan-  iliis. 

11 

29 

56 

49 

11 

18 

51 

966 

961 

972 

966 

964 

995 

_  ,  C  Com. 
Feb-  iBis. 

0 

1 

38 

30 

11 

15 

43 

54 

224 

875 

45 

111 

165 

0 

1 

35 

19 

11 

4 

34 

20 

185 

847 

11 

75 

159 

March,  - 

0 

3 

7 

27 

9 

27 

59 

7 

330 

666 

989 

114 

313 

April,  -  - 

0 

4 

45 

57  1 

9 

13 

42 

61 

554 

542 

34 

225 

478 

May,  -  -  - 

0 

6 

21 

16 

8 

18 

15 

81 

738 

389 

46 

300 

638 

June,  -  - 

0 

7 

59 

46 

8 

3 

58 

136 

962 

264 

91 

411 

802 

July,  -  -  - 

0 

9 

35 

5 

7 

8 

32 

156 

147 

112 

103 

486 

962 

August,  - 

0 

11 

13 

35 

6 

24 

15 

210 

371 

987 

147 

597 

126 

September, 

0 

12 

52 

5 

6 

9 

58 

265 

595 

862 

193 

708 

291 

October,  - 

0 

14 

27 

24 

5 

14 

32 

285 

780 

710 

204 

783 

451 

November, 

0 

16 

5 

53 

5 

0 

15 

339 

4 

585 

250 

894 

615 

December, 

0 

17 

41 

13  : 

4 

4 

49 

359 

188 

432 

261  1 

969 

775 

26 


TABLE  XXIL 


Moon's  Motions  for  Days. 


Days 

1 

2 

3 

4  |. 

5 

6 

7 

8 

9 

1 

3000 

0000 

0000 

0000 

0000 

0000 

0000 

0000  ( 

)000 

2 

27 

650 

1040 

287 

336 

372 

58 

390 

24 

3 

55 

1300 

2080 

574 

671  ' 

■  744 

115 

781 

49 

4 

82 

1950 

3121 

861 

1007 

1116 

173 

1171 

73 

5 

109 

2600 

4161 

1148 

1342 

1488 

231 

1561, 

97 

6 

137 

3249 

5201 

1435 

1678 

1860 

289 

1952 

121 

7 

164 

3899 

6241 

1722 

2013 

2232 

346 

2342 

146 

8 

192 

4549 

7281 

2009 

2349 

2604 

404 

2732 

170 

9 

219 

5199 

8321 

2296 

2684 

2976 

462 

3122 

194 

10 

246 

5849 

9362 

2583 

3020 

3348 

519 

3513 

219 

11 

274 

6499 

402 

2870 

oo55 

3720 

577 

3903 

243 

12 

301 

7149 

1442 

3157 

3691 

4093 

635 

4293 

267 

13 

328 

7799 

2482 

3444 

4026 

4465 

692 

4684 

291 

14 

356 

8449 

3522 

3731 

4362 

4837 

750 

5074 

316 

15 

383 

9098 

4563 

4018 

4698 

5209 

808 

5464 

340 

16 

411 

9748 

5603 

4305 

5033 

5581 

866 

5854 

364 

17 

438 

398 

6643 

4592 

5369 

5953 

923 

6245 

389 

18 

465 

1048 

7683 

4878 

5704 

6325 

981 

6635 

413 

19 

493 

1698 

8723 

5165 

6040 

6697 

1039 

7025 

437 

20 

520 

2348 

9763 

5452 

6375 

7069 

1096 

7416 

461 

21 

548 

2998 

804 

5739 

6711 

7441 

1154 

7806 

486 

22 

575 

3648 

1844 

6026 

7046 

7813 

1212 

8196 

510 

23 

602 

4298 

2884 

6313 

7382 

8185 

1269 

8586 

534 

24 

630 

4947 

3924 

6600 

7717 

8557 

1327 

8977 

559 

25 

657 

5597 

4964 

6887 

8053 

8929 

1385 

9367 

583 

26 

684 

6247 

6005 

7174 

8389 

9301 

1443 

9757 

607 

27 

712 

6897 

7045 

7461 

8724 

9673 

1500 

148. 

631 

28 

739 

7547 

8085 

7748 

9060 

45 

1558 

538 

656 

29 

767 

8197 

9125 

8035 

9395 

417 

1616 

928 

680 

30 

794 

8847 

165 

8322 

9731 

789 

1673 

1319 

704 

31 

821 

9497 

1205 

8609 

66 

1161 

1731 

1709 

729 

TABLE  XXIJ 


27 


Moon's  Motions  for  Days. 


Days 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

1 

000 

000 

000 

000 

000 

000 

000 

000 

000 

000 

000 

2 

70 

31 

70 

34 

99 

31 

37 

42 

26 

0 

0 

3 

140 

62 

141 

68 

198 

61 

73 

84 

52 

0 

1 

4 

210 

93 

211 

103 

297 

92 

110 

126 

78 

0 

1 

5 

281 

125 

282 

137 

397 

122 

146 

168 

104 

1 

2 

6 

351 

156 

352 

171 

496 

153 

183 

210 

130 

1 

2 

7 

421 

187 

423 

205 

595 

183 

220 

252 

156 

1 

3 

8 

491 

218 

493 

239 

694 

214 

256 

294 

182 

1 

3 

9 

561 

249 

564 

273 

793 

244 

293 

336 

208 

1 

4 

10 

631 

280 

634 

308 

892 

275 

329 

379 

234 

1 

4 

11 

702 

311 

705 

342 

992 

305 

366 

421 

260 

1 

5 

12 

772 

342 

775 

376 

91 

336 

403 

463 

286 

2 

5 

13 

842 

374 

845 

410 

190 

366 

439 

505 

312 

2 

5 

14 

912 

405 

916 

444 

289 

397 

476 

547 

337 

2 

6 

15 

982 

436 

986 

478 

388 

427 

512 

589 

363 

2 

6 

16 

52 

467 

57 

513 

487 

458 

549 

631 

389 

2 

7 

17 

122 

498 

127 

547 

587 

488 

586 

673 

415 

2 

7 

18 

193 

529 

198 

581 

686 

519 

622 

715 

441 

2 

8 

19 

263 

560 

268 

615 

785 

549 

659 

757 

467 

3 

8 

20 

333 

591 

339 

649 

884 

580 

695 

799 

493 

3 

9 

21 

403 

623 

409 

683 

983 

611 

732 

841 

519 

3 

9 

22 

473 

654 

480 

718 

82 

641 

769 

883 

545 

3 

10 

23 

543 

685 

550 

752 

182 

672 

805 

925 

571 

3 

10 

24 

614 

716 

621 

786 

281 

702 

842 

967 

597 

3 

11 

25 

684 

747 

691 

820 

380 

733 

878 

9 

623 

4 

11 

26 

754 

778 

762 

854 

479 

763 

915 

52 

649 

4 

11 

27 

824 

809 

832 

888 

578 

794 

952 

94 

675 

4 

12 

28 

894 

840 

903 

923 

677 

824 

988 

136 

701 

4 

12 

29 

964 

872 

973 

957 

777 

855 

25 

178 

727 

4 

13 

30 

34 

903 

43 

991 

876 

885 

61 

220 

753 

4 

13 

31 

105 

934 

114 

25 

975 

916 

98 

262 

779 

4 

14 

38 


TABLE  XXII. 


Moon's  Motions  for  Days. 


Days 

Evection. 

Anomaly. 

Variation. 

M.  Long. 

1 

Os  o°  0'  0" 

0s  0°  0'  0" 

0*  0°  O'  0" 

0*  0°  0'  0" 

2 

0  11  18  59 

0  13  3  54 

0  12  11  27 

0  13  10  35 

3 

0  22  37  59 

0  26  7  48 

0  24  22  53 

0  26  21  10 

4 

1  3  56  58 

1  9  11  42 

1  6  34  20 

1  9  31  45 

5 

1  15  15  58 

1  22  15  36 

1  18  45  47 

1  22  42  20 

6 

1  26  34  57 

2  5  19  30 

2  0  57  13 

2  5  52  55 

7 

2  7  53  57 

2  18  23  24 

2  13  8  40 

2  19  3  30 

8 

2  19  12  56 

3  1  27  18 

2  25  20  7 

3  2  14  5 

9 

3  0  31  55 

3  14  31  12 

3  7  31  34 

3  15  24  40 

10 

3  11  50  55 

3  27  35  6 

3  19  43  0 

3  28  35  15 

11 

3  23  9  54 

4  10  39  0 

4  154  27 

4  11  45  50 

12 

4  4  28  54 

4  23  42  54 

4  14  5  54 

4  24  56  25 

13 

4  15  47  53 

5  6  46  48 

4  26  17  20 

5  8  7  0 

14 

4  27  6  53 

5  19  50  42 

5  8  28  47 

5  21  17  35 

15 

5  8  25  52 

6  2  54  36 

5  20  40  14 

6  4  28  10 

16 

5  19  44  51 

6  15  58  29 

6  2  51  40 

6  17  38  45 

17 

6  1  3  51 

6  29  2  23 

6  15  3  7 

7  0  49  20 

18 

6  12  22  50 

7  12  6  17 

6  27  14  34 

7  13  59  55 

19 

6  23  41  50 

7  25  10  11 

7  9  26  1 

7  27  10  30 

20 

7  5  0  49 

8  8  14  5 

7  21  37  27 

8  10  21  5 

21 

7  16  19  49 

8  21  17  59 

8  3  48  54 

8  23  31  40 

22 

7  27  38  48 

9  4  21  53 

8  16  0  21 

9  6  42  16 

23 

8  8  57  47 

9  17  25  47 

8  28  11  47 

9  19  52  51 

24 

8  20  16  47 

10  0  29  41 

9  10  23  14 

10  3  3  26 

25 

9  1  35  46 

10  13  33  35 

9  22  34  41 

10  16  14  1 

26 

9  12  54  46 

10  26  37  29 

10  4  46  7 

10  29  24  36 

27 

9  24  13  45 

11  9  41  23 

10  16  57  34 

11  12  35  11 

28 

10  5  32  45 

11  22  45  17 

10  29  9  1 

11  25  45  46 

29 

10  16  51  44 

0  5  49  11 

11  11  20  28 

0  8  56  21 

30 

10  28  10  43 

0  18  53  5 

11  23  31  54 

0  22  6  56 

31 

11  9  29  43 

1  1  56  59 

0  5  43  21 

1  5  17  31 

TARLE  XXII 


29 


Moon's  Motions  for  Days. 


ill 

Sup 

vOt 

Node. 

II. 

V. 

VI. 

VIA. 

via. 

IX. 

! x' 

1 

0°  0' 

0" 

0* 

0°  O' 

000 

000 

000 

000 

000 

000 

2 

0 

3 

11 

11 

9 

34 

39 

28 

34 

36 

5 

3 

0 

6 

21 

22 

18 

68 

79 

56 

67 

72 

11 

4 

0 

9 

32 

1 

3 

27 

102 

118 

85 

101 

108 

16 

5 

0 

12 

42 

1 

14 

37 

136 

158 

113 

135 

143 

21 

6 

0 

15 

53 

1 

25 

46 

170 

197 

141 

169 

179 

27 

7 

0 

19 

4 

2 

6 

55 

204 

237 

169 

202 

215 

32 

8 

0 

22 

14 

2 

18 

4 

238 

276 

198 

236 

251 

37 

9 

0 

25 

25 

2 

29 

13 

272 

316 

226 

270 

287 

43 

10 

0 

28 

36 

3 

10 

22 

306 

355 

254 

303 

323 

48 

11 

0 

31 

46 

3 

21 

31 

340 

395 

282 

337 

358 

53 

12 

0 

34 

57 

4 

2 

40 

374 

434 

311 

371 

394 

58 

13 

0 

38 

8 

4 

13 

50 

408 

474 

339 

405 

430 

64 

14 

0 

41 

18 

4 

24 

59 

442 

513 

367 

438 

466 

69 

15 

0 

44 

29 

5 

6 

8 

476 

553 

395 

472 

502 

74 

16 

0 

47 

39 

5 

17 

17 

510 

592 

424 

506 

538 

80 

17 

0 

50 

50 

5 

28 

26 

544 

632 

452 

539 

573 

85 

18 

0 

54 

1 

6 

9 

35 

578 

671 

480 

573 

609 

90 

19 

0 

57 

11 

6 

20 

44 

612 

711 

508 

607 

645 

96 

20 

1 

0 

22 

7 

1 

53 

646 

750 

537 

641 

681 

101 

21 

1 

o 

o 

33 

7 

13 

3 

680 

790 

565 

674 

717 

106 

22 

1 

6 

43 

7 

24 

12 

714 

829 

593 

708 

753 

112 

23 

1 

9 

54 

8 

5 

21 

748 

869 

621 

742 

788 

117 

24 

1 

13 

5 

8 

16 

30 

782 

908 

650 

775 

824 

122 

25 

1 

16 

15 

8 

27 

39 

816 

948 

678 

809 

860 

128 

26 

1 

19 

26 

9 

8 

48 

850 

987 

706 

843 

896 

133 

27 

1 

22 

36 

9 

19 

57 

884 

027 

734 

877 

932 

138 

28 

1 

25 

47 

10 

1 

6 

918 

066 

762 

910 

968 

143 

29 

1 

28 

58 

10 

12 

16 

952 

106 

791 

944 

003 

149 

30 

1 

32 

8 

10 

23 

25 

986 

145 

819 

978 

039 

154  ! 

31 

1 

35 

19 

11 

4 

34 

020 

185 

847  | 

011 

075 

159 

30 


TABLE  XXIII. 


Moon's  Motions  for  Hours. 


Hours. 

1 

2 

3 

4 

5 

6 

1  7 

8 

9 

1 

1 

27 

43 

12 

14  ' 

16 

2 

16 

1 

2 

2 

54 

87 

24 

28 

31 

5 

33 

2 

3 

3 

81 

130 

36 

42 

47 

7 

49 

3 

4 

5 

108 

173 

48 

56 

62 

10 

65 

4 

5 

6 

135 

217 

60 

70 

78 

12 

81 

5 

6 

7 

162 

260 

72 

84 

93 

14 

98 

6 

7 

8 

190 

303 

84 

98 

109 

17 

114 

7 

8 

9 

217 

347 

96 

112 

124 

19 

130 

8 

9 

10 

244 

390 

108 

126 

140 

22 

146 

9 

10 

11 

271 

433 

120 

140 

155 

24 

163 

10 

11 

12 

298 

477 

131 

154 

171 

26 

179 

11 

12 

14 

325 

520 

143 

168 

186 

29 

195 

12 

13 

15 

352 

563 

155 

182 

202 

31 

211 

13 

14 

16 

379 

607 

167 

196 

217 

34 

228 

14 

15 

17 

406 

650 

179 

210 

233 

36 

244 

15 

16 

18 

433 

693 

191 

224 

248 

38 

260 

16 

17 

19 

460 

737 

203 

238 

264 

41 

276 

17 

18 

20 

487 

780 

215 

252 

279 

43 

293 

18 

19 

22 

515 

823 

227 

266 

295 

46 

309 

19 

20 

23 

542 

867 

239 

280 

310 

48 

325 

20  4 

21 

24 

569 

910 

251 

294 

326 

50 

341 

21 

22 

25 

596 

953 

263 

308 

341 

53 

358 

22 

23 

26 

623 

997 

275 

322 

357 

55 

374 

23 

24 

27 

650 

1040 

287 

336 

372 

58 

390 

24 

TAfiLE  XXIII 


Moon's  Motions  for  Hours . 


Hours. 

10 

11 

12 

13 

14 

15 

16 

17 

a 

1 

3 

1 

3 

1 

4 

1 

2 

2 

1 

2 

6 

3 

6 

3 

8 

3 

3 

4 

2 

3 

9 

4 

9 

4 

12 

4 

5 

5 

3 

1 

4 

12 

5 

12 

6 

16 

5 

6 

7 

4  ! 

5 

15 

6 

15 

7 

21 

6 

8 

9 

5 

6 

18 

8 

18 

9 

25 

8 

9 

11 

6 

7 

20 

9 

20 

10 

29 

9 

11 

12 

8  ", 

8 

23 

10 

23 

11 

33 

10 

12 

14 

9 

9 

26 

12 

26 

13 

37 

11 

14 

16 

10 

10 

29 

13 

29 

14 

41 

13 

15 

18 

11 

11 

32 

14 

32 

16 

45 

14 

17 

19 

12 

12 

35 

16 

35 

17 

49 

15 

18 

21 

13 

13 

38 

17 

38 

18 

54 

16 

20 

23 

14 

14 

41 

18 

41 

20 

58 

18 

21 

25 

15 

15 

44 

19 

44 

21 

62 

19 

23 

26 

16 

16 

47 

21 

47 

23 

66 

20 

25 

28 

17 

17 

50 

22 

50 

24 

70 

21 

26 

30 

18 

18 

53 

23 

53 

25 

74 

23 

28 

32 

19 

19 

56 

25 

56 

27' 

78 

24 

29 

33. 

21 

20 

58 

26 

58 

28 

83 

25 

31 

35 

22 

21 

61 

27 

61 

30 

87 

26 

32 

37 

23 

22 

64 

28 

64 

31 

91 

28 

34 

39 

24 

23 

67 

30 

67 

33 

95 

29 

35 

40 

25 

24 

70 

1  31 

70 

34 

99 

31 

37 

42 

26  ! 

32 


TABLE  XXIII, 


Moon’s  Motions  for  Hours . 


Hours. 

Evection. 

Anomaly. 

Variation. 

Longitude. 

1 

0°  28'  17" 

0°  32'  40" 

'  0°  30'  29" 

'  0°  32'  56" 

2 

0  56  35 

1  5  19 

1  0  57 

1  5  53 

3 

1  24  52 

1  37  59 

1  31  26 

1  38  49 

4 

1  53  10 

2  10  39 

2  1  54 

2  11  46 

5 

2  21  27 

2  43  19 

2  32  23 

2  44  42 

6 

2  49  45 

3  15  58 

3  2  52 

3  17  39 

7 

3  18  2 

3  48  38 

3  33  20 

3  50  35 

8 

3  46  20 

4  21  18 

4  3  49 

4  23  32 

9 

4  14  37 

4  53  58 

4  34  17 

4  56  28 

10 

4  42  55 

5  26  37 

5  4  46 

5  29  25 

11 

5  11  12 

5  59  17 

5  35  15 

6  2  21 

12  , 

5  39  30 

6  31  57 

6  5  43 

6  35  17 

13 

6  7  47 

7  4  37 

6  36  12 

7  8  14 

14 

6  36  5 

7  37  16 

7  6  40 

7  41  10 

15 

7  4  22 

8  9  56 

7  37  9 

8  14  7 

16 

7  32  40 

8  42  36 

8  7  38 

8  47  3 

17 

8  0  57 

9  15  16 

8  38  6 

9  20  0 

18 

8  29  15 

9  47  55 

9  8  35 

9  52  56 

19 

8  57  32 

10  20  35 

9  39  3 

10  25  53 

20 

9  25  50 

10  53  15 

10  9  32 

10  58  49 

21 

9  54  7 

11  25  55 

10  40  1 

11  31  46 

22 

10  22  24 

11  58  34 

11  10  29 

12  4  42 

23 

10  50  42 

12  31  14 

11  40  58 

12  37  39 

24 

11  18  59 

13  3  54 

12  11  27 

13  10  35  j 

TABLE  XXIU. 

Moon's  Motions  for  Hours . 


Hours, 

Sup.  of 
Node. 

II. 

V. 

VI. 

VII. 

VIII 

IX. 

X. 

j  1 

0' 

8" 

0° 

28' 

1 

2 

1 

1 

1 

0 

i  2 

0 

16 

0 

56 

3 

3 

2 

3 

3 

0 

[  3 

0 

24 

1 

24 

4 

5 

4 

4 

4 

1 

1  4 

0 

32 

1 

52 

6 

7 

5 

6 

6 

1 

;  5 

0 

40 

2 

19 

7 

8 

6 

7 

7 

1 

|  6 

0 

48 

2 

47 

9 

10 

7 

9 

9 

1 

j  7 

0 

56 

3 

15 

10 

12 

8 

10 

10 

2 

8 

1 

4 

3 

43 

11 

13 

9 

11 

12 

2 

9 

1 

11 

4 

11 

13 

15 

11 

13 

13 

2 

1  io 

1 

19 

4 

39 

14 

16 

12 

14 

15 

2 

11 

1 

27 

5 

7 

16 

18 

13 

15 

16 

2 

1  12 

1 

1 

35 

5 

35 

17 

20 

14 

17 

18 

3 

f  13 

1 

43 

6 

2 

18  ' 

21 

15 

18 

19 

3 

•  14 

1 

51 

6 

30 

20 

23 

16 

19 

21 

3 

1 

59 

6 

58 

21 

25 

18 

21 

22 

3 

16 

2 

7 

7 

26 

23 

26 

19 

22 

24 

4 

17 

2 

15 

7 

54 

24 

28 

20 

24 

25 

4 

18 

2 

23 

8 

22 

26 

29 

21 

25 

27 

4 

19 

2 

31 

8 

50 

27 

31 

22 

27 

28 

4 

20 

2 

39 

9 

18 

28 

32 

24 

28 

30 

4 

21 

2 

47 

9 

45 

30 

34 

25 

29 

31 

5 

!  22 

2 

55 

10 

13 

31 

36 

26 

31 

33 

5 

23 

3 

3 

10 

41 

33 

38 

27 

32 

34 

5 

!  24 

3 

11 

11 

9 

1  34 

39 

28 

34 

36 

5 

3* 


33 


f 


34?  TABLE  XXIV. 


Moon's  Motions  for  Minutes  and  Seconds . 


Min. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

1 

0 

0 

1 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

2 

0 

1 

1 

0 

0 

1 

0 

1 

0 

0 

0 

0 

0 

0 

3 

0 

1 

2 

1 

1 

1 

0 

1 

0 

0 

0 

0 

0 

0 

4 

0 

2 

3 

1 

1 

1 

0 

1 

0 

0 

0 

0 

0 

0 

5 

0 

2 

4 

1 

1 

1 

0 

1 

0 

0 

0 

0 

0 

0 

6 

0 

3 

4 

1 

1 

2 

0 

2 

0 

0 

0 

0 

0 

0 

7 

0 

3 

5 

1 

2 

2 

0 

2 

0 

0 

0 

0 

0 

0 

8 

0 

4 

6 

2 

2 

2 

0 

2 

0 

0 

0 

0 

0 

1 

9 

0 

4 

6 

2 

2 

2 

0 

2 

0 

0 

0 

0 

0 

1 

10 

0 

5 

7 

2 

2 

3 

0 

3 

0 

0 

0 

0 

0 

1 

11 

0 

5 

8 

2 

3 

3 

0 

3 

0 

1 

0 

1 

0 

1 

12 

0 

5 

9 

2 

3 

3 

0 

3 

0 

1 

0 

1 

0 

1 

13 

0 

6 

9 

3 

3 

3 

1 

4 

0 

1 

0 

1 

0 

1 

14 

0 

6 

10 

3 

3 

4 

1 

4 

0 

1 

0 

1 

0 

1 

15 

0 

7 

11 

3 

3 

4 

1 

4 

0 

1 

0 

1 

0 

1 

16 

0 

7 

12 

3 

4 

4 

1 

4 

0 

1 

0 

1 

0 

1 

17 

0 

8 

12 

3 

4 

4 

1 

5 

0 

1 

0 

1 

0 

1 

18 

0 

8 

13 

4 

4 

5 

1 

5 

0 

1 

0 

1 

0 

1 

19 

0 

9 

14 

4 

4 

5 

1 

5 

0 

1 

0 

1 

0 

1 

20 

0 

9 

14 

4 

5 

5 

1 

5 

0 

1 

0 

1 

0 

1 

21 

0 

10 

15 

4 

5 

5 

1 

6 

0 

1 

0 

1 

0 

1 

22 

0 

10 

16 

4 

5 

6 

1 

6 

0 

1 

0 

1 

1 

2 

23 

0 

10 

17 

5 

5 

6 

1 

6 

0 

1 

0 

1 

1 

2 

24 

0 

11 

17 

5 

6 

6 

1 

7 

0 

1 

1 

1 

1 

2 

25 

0 

11 

18 

5 

6 

6 

1 

7 

0 

1 

1 

1 

1 

2 

26 

0 

12 

19 

5 

6 

7 

1 

7 

0 

1 

1 

1 

1 

2 

27 

1 

12 

19 

5 

6 

7 

1 

7 

0 

1 

1 

1 

1 

2 

28 

1 

13 

20 

6 

7 

7 

1 

8 

0 

1 

1 

1 

1 

2 

29 

1 

13 

21 

6 

7 

7 

1 

8 

0 

1 

1 

1 

1 

2 

30 

1 

14 

22 

6 

7 

8 

1 

8 

0 

1 

1 

1 

1 

2 

i 


TABLE  XXIV, 


3£ 


Moon's  Motions  for  Minutes  and  Seconds. 


Min. 

Evec. 

Anom. 

Variat. 

Long. 

Sup. 

nod. 

II. 

Sec. 

Ev. 

An. 

Var. 

Lon. 

1 

O' 28" 

0'  33" 

0'  30" 

0'  33' 

0" 

0' 

1 

0" 

0" 

1" 

0" 

2 

0  57 

1  5 

1  1 

1  6 

0 

1 

2 

1 

1 

1 

1 

3 

1  25 

1  38 

1  31 

1  39 

0 

1 

3 

1 

2 

2 

2 

4 

1  53 

2  11 

2  2 

2  12 

0 

2 

4 

2 

2 

2 

2 

5 

2  21 

2  43 

2  32 

2  45 

1 

2 

5 

2 

3 

3 

3 

6 

2  50 

3  16 

3  3 

3  18 

1 

3 

6 

3 

3 

3 

3 

7 

3  18 

3  49 

3  33 

3  51 

1 

3 

7 

3 

4 

4 

4 

8 

3  46 

4  21 

4  4 

4  23 

1 

4 

8 

4 

4 

4 

4 

9 

4  15 

4  54 

4  34 

4  56 

1 

4 

9 

4 

5 

5 

5 

10 

4  43 

5  27 

5  5 

5  29 

1 

5 

10 

5 

5 

5 

5 

11 

5  11 

5  59 

5  35 

6  2 

1 

5 

11 

5 

6 

6 

6 

12 

5  40 

6  32 

6  6 

6  35 

2 

6 

12 

6 

6 

6 

7 

13 

6  8 

7  5 

6  36 

7  8 

2 

6 

13 

6 

7 

7 

7 

14 

6  36 

7  37 

7  7 

7  41 

2 

7 

14 

7 

8 

7 

8 

15 

7  4 

8  10 

7  37 

8  14 

2 

7 

15 

7 

8 

8 

8 

16 

7  33 

8  43 

8  8 

8  47 

2 

7 

16 

8 

9 

8 

9 

17 

8  1 

9  15 

8  38 

9  20 

2 

8 

17 

8 

9 

9 

9 

18 

8  29 

9  48 

9  9 

9  53 

2 

8 

18 

9 

10 

9 

10 

19 

8  58 

10  21 

9  39 

10  26 

2 

9 

19 

9 

10 

10 

10 

20 

9  26 

10  53 

10  10 

10  59 

3 

9 

20 

9 

11 

10 

11 

21 

9  54 

11  26 

10  40 

11  32 

3 

10 

21 

10 

11 

11 

11 

22 

10  22 

11  59 

11  11 

12  5 

3 

10 

22 

10 

12 

11 

12 

23 

10  51 

12  31 

11  41 

12  38 

3 

11 

23 

11 

12 

12 

13 

24 

11  19 

13  4 

12  12 

13  11 

3 

11 

24 

11 

13 

12 

13 

25 

11  47 

13  37 

12  42 

13  43 

3 

12 

25 

12 

14 

13 

14 

26 

12  16 

14  9 

13  13 

14  16 

3 

12 

26 

12 

14 

13 

14 

27 

12  44 

14  42 

13  43 

14  49 

4 

13 

27 

13 

15 

14 

15 

28 

13  12 

15  15 

14  13 

15  22 

4 

13 

28 

13 

15 

14 

15 

29 

13  40 

15  47 

14  44 

15  55 

4 

13 

29 

14 

16 

15 

16 

30 

14  9 

16  20 

15  14 

16  28 

4 

14 

30 

14 

16 

15 

16 

86 


TABLE  XXIV 


Moon's  Motions  for  Minutes  and  Seconds. 


Min. 

1 

2  ' 

O 

O 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

31 

1 

14 

22 

6 

7 

8 

1 

8 

0 

1 

1 

1 

1 

2 

32 

1 

14 

23 

6 

7 

8 

1 

9 

1 

2 

1 

2 

1 

2 

33 

1 

15 

24 

7 

8 

9 

1 

9 

1 

2 

1 

2 

1 

2 

34 

1 

15 

25 

7 

8 

9 

1 

9 

1 

2 

1 

2 

1 

2 

35 

1 

16 

25 

7 

8 

9 

1 

10 

1 

2 

1 

2 

1 

2 

36 

1 

16 

26 

7 

8 

9 

1 

10 

1 

2 

1 

2 

1 

3 

37 

1 

17 

27 

7 

9 

10 

1 

10 

1 

2 

1 

2 

1 

3 

38 

1 

17 

27 

8 

9 

10 

2 

10 

1 

2 

1 

2 

1 

3 

39 

1 

18 

28 

8 

9 

10 

2 

11 

1 

2 

1 

2 

1 

3 

40 

1 

18 

29 

8 

9 

10 

2 

11 

1 

2 

1 

2 

1 

3 

41 

1 

19 

30 

8 

10 

11 

2 

11 

1 

2 

1 

2 

1 

3 

42 

1 

19 

30 

8 

10 

11 

2 

11 

1 

2 

1 

2 

1 

3 

43 

1 

19 

31 

9 

10 

11 

2 

12 

1 

2 

1 

2 

1 

3 

44 

1 

20 

32 

9 

10 

11 

2 

12 

1 

2 

1 

2 

1 

3 

45 

1 

20 

32 

9 

10 

12 

2 

12 

1 

2 

1 

2 

1 

3 

46 

1 

21 

33 

9 

11 

12 

2 

12 

1 

2 

1 

2 

1 

3 

4  7 

1 

21 

34 

9 

11 

12 

2 

13 

1 

2 

1 

2 

1 

3 

48 

1 

22 

35 

10 

11 

12 

2  ; 

i  13 

1 

2 

1 

2 

1 

3 

49 

1 

22 

35 

10 

11 

13 

2 

13 

1 

2 

1 

2 

1 

3 

50 

1 

23 

36 

10 

11 

13 

2 

13 

1 

2 

1 

2 

1 

3 

51 

1 

23 

37 

10 

12 

13 

2 

14 

1 

2 

1 

2 

1 

4 

52 

1 

24 

38 

10 

12 

13 

2 

14 

1 

3 

1 

3 

1 

4 

53 

1 

24 

38 

11 

12 

14 

2 

14 

1 

3 

1 

3 

1 

4 

54 

1 

24 

39 

11 

12 

14 

2 

14 

1 

3 

1 

3 

1 

4 

55 

1 

25 

40 

11 

13 

14 

2 

15 

1 

3 

,  1 

3 

1 

4 

56 

1 

25 

40 

11 

13 

14 

2 

15 

1 

3 

1 

3 

1 

4 

57 

1 

26 

41 

11 

13 

15 

2 

15 

1 

3 

1 

3 

1 

4 

58 

1 

26 

42 

12 

13 

15 

2 

16 

1 

3 

1 

3 

1 

4 

59 

1 

27 

43 

12 

14 

15 

2 

16 

1 

3 

1 

3 

1 

4 

60 

1 

27 

43 

12 

14 

15 

2 

16 

1 

o 

O 

1 

3 

1 

4 

TABLE  XXIV 


3  7 


Moon's  Motions  for  Minutes  and  Seconds. 


Min. 

Evect. 

Anom. 

Variut. 

Long1. 

Sup. 

nod. 

n. 

Sec. 

Ev. 

An. 

Var.' 

Lon. 

31 

14'  37" 

16' 52" 

15' 45" 

17'  1" 

4" 

14' 

31 

15" 

17" 

16" 

17" 

32 

15  5 

17  25 

16  15 

17  34 

4 

15 

32 

15 

17 

16 

18 

33 

15  34 

17  58 

16  46 

18  7 

4 

15 

33 

16 

18 

17 

18 

34 

16  2 

18  30 

17  16 

18  40 

4 

16 

34 

16 

18 

17 

19 

35 

16  30 

19  3 

17  47 

19  13 

5 

16 

35 

17 

19 

18 

19 

36 

16  58 

19  36 

18  17 

19  46 

5 

17 

36 

17 

20 

18 

20 

37 

17  27 

20  8 

18  48 

20  19 

5 

17 

37 

18 

20 

19 

20 

38 

17  55 

20  41 

19  18 

20  52 

5 

18 

38 

18 

21 

19 

21 

39 

18  23 

21  14 

19  49 

21  25 

5 

18 

39 

18 

21 

20 

21 

40 

18  52 

21  46 

20  19 

21  58 

5 

19 

40 

19 

22 

20 

22 

41 

19  20 

22  19 

20  50 

22  31 

5 

19 

41 

19 

22 

21 

22 

42 

19  48 

22  52 

21  20 

23  3 

6 

20 

42 

20 

23 

21 

23 

43 

20  16 

23  24 

21  51 

23  36 

6 

20 

43 

20 

23 

22 

24 

44 

20  45 

23  57 

22  21 

24  9 

6 

21 

44 

21 

24 

22 

24 

45 

21  13 

24  30 

22  52 

24  42 

6 

21 

45 

21 

24 

23 

25 

46 

21  41 

25  2 

23  22 

25  15 

6 

21 

46 

22 

25 

23 

25 

47 

22  10 

25  35 

23  53 

25  48 

6 

22 

47 

22 

26 

24 

26 

48 

22  38 

26  8 

24  23 

26  21 

6 

22 

48 

23 

26 

24 

26 

49 

23  6 

26  40 

24  54 

26  54 

6 

23 

49 

23 

27 

25 

27 

50 

23  34 

27  13 

25  24 

27  27 

7 

23 

50- 

24 

27 

25 

27 

51 

24  3 

27  46 

25  55 

28  0 

7 

24 

51 

24 

28 

26 

28 

52 

24  31 

28  18 

26  25 

28  33 

7 

24 

52 

25- 

28 

26 

28 

53 

24  59 

28  51 

26  56 

29  6 

7 

25 

53 

25 

29 

27 

29 

54 

25  28 

29  24 

27  26 

29  39 

7 

25 

54 

26 

29 

27 

30 

55 

25  56 

29  56 

27  56 

30  12 

7 

26 

55 

26 

30 

28 

30 

56 

26  24 

30  29 

28  27 

30  45 

7 

26 

56 

26 

30 

28 

31 

57 

26  52 

31-  2 

28  57 

31  18 

7 

27 

57 

27 

31 

29 

31 

58 

27  21 

31  34 

29  28 

31  51 

8 

27 

58 

27 

.32 

29 

32 

59 

27  49 

32  7 

29  58 

32  23 

8 

28 

59 

28 

32 

30 

32 

60 

28  17 

32  40 

30  29 

32  56 

8 

28 

■60 

28 

33 

30 

33 

38 


TABLE  XXV\ 


First  Equation  of  Moon's  Longitude.  Argument  1, 


Arg. 

1 

Uiff. 

Arg. 

1 

Diff*. 

0 

12' 40" 

42" 

42 

42 

41 

41 

40 

5000 

12'  40" 

40". 

41 

40 

39 

40 
38 

100 

11  58 

5100 

13  20 

200 

11  16 

5200 

14  1 

300 

10  34 

5300 

14  41 

400 

9  53 

5400 

15  20 

500 

9  12 

5500 

16  0 

600 

8  32 

38 

38 

36 

34 

33 

5600 

16  38 

37 
37  ' 
35 
34 
32 

700 

7  54 

5700 

17  15 

800 

7  16 

5800 

17  52 

900 

6  40 

5900 

18  27 

1000 

6  6 

6000 

19  1 

1100 

5  33 

31 

30 

27 

25 

23 

6100 

19  33 

31 

29 

28 

26 

23 

1200 

5  2 

6200 

20  4 

1300 

4  32 

6300 

20  33 

1400 

4  5 

6400 

21  1 

1500 

3  40 

6500 

21  27 

1600 

3  17 

21 

18 

16 

13 

11 

6600 

21  50 

22 

19 

17 

15 

12 

1700 

2  56 

6700 

22  12 

1800 

2  38 

6800 

22  31 

1900 

2  22 

6900 

22  48 

2000 

2  9 

7000 

23  3 

2100 

1  58 

8 

7100 

23  15 

10 

* 7 

2200 

1  50 

7200 

23  25 

2300 

1  44 

o 

Q 

7300 

23  32 

t 

2400 

1  41 

O 

0 

7400 

23  37 

O 

O 

2500  ; 

|  1  41 

2 

7500 

23  39 

z 

0 

2600 

1  43 

€ 

7600 

23  39 

<3 

2700 

1  48 

O 

7 

7700 

23  36 

o 

f i 

2800 

1  55 

10 

12 

15 

7800 

23  30 

u 

8 

11 

13 

2900 

2  5 

7900 

23  22 

3000 

2  17 

8000 

23  11 

3100 

2  32 

17 

19 

22 

23 

26 

8100 

22  58 

16 

18 

21 

23 

25 

3200 

2  49 

8200 

22  42 

3300 

3  8 

8300 

22  24 

3400 

3  30 

8400 

22  3 

3500 

3  53 

8500 

21  40 

3600 

4  19 

27 

30 

^  1 

8600 

21  15 

27 

30. 

31 

3700 

4  46 

8700 

20  48 

3800 

5  16 

8800 

20  18 

3900 

5  47 

O  1 

32 

34 

8900 

19  47 

4000 

6  19 

9000 

19  14 

oo 

34 

4100 

6  53 

9100 

18  40 

36 

38 

38 

40 

41 

4200 

7  28 

oo 

37 

37 

38 

39 

9200 

18  4 

4300 

8  5 

9300 

17  26 

4400 

8  42 

9400 

16  48 

4500 

9  20 

9500 

16  8 

4600 

9  59 

40 

40 

40 

41 

9600 

15  27 

41 

42 
42 
42 

4700 

10  39 

9700 

14  46 

4800 

11  19 

9800 

14  4 

4900 

11  59 

9900 

13  22 

5000 

12  40 

10000 

12  40 

TABLE  XXVL 


Equations  2  to!  of  Moon's  Longitude.  Arguments  2  to  7. 


Arg. 

2 

3 

4 

5 

6 

7 

Arg. 

2500 

4'  57" 

O'  2" 

6' 30" 

3' 39" 

0'  6" 

O'  1" 

2500 

2600 

4  57 

0  2 

6  30 

3  39 

0  6 

0  1 

2*400 

2700 

4  56 

0  3 

6  29 

3  38 

0  7 

0  1 

2300 

2800 

4  55 

0  3 

6  27 

3  37 

0  8 

0  2 

2200 

2900 

4  53 

0  4 

6  24 

3  36 

0  9 

0  3 

2100 

3000 

4  50 

0  5 

6  21 

3  34 

0  10 

0  4 

2000 

3100 

4  47 

0  6 

6  17 

3  32 

0  12 

0  5 

1900 

3200 

4  43 

0  8 

6  12 

3  29 

0  14 

0  6 

1800 

3300 

4  39 

0  9 

6  7 

3  26 

0  17 

0  8 

1700 

3400 

4  34 

0  11 

6  1 

3  22 

0  19 

0  10 

1600 

3500 

4  29 

0  13 

5  54 

3  18 

0  22 

0  12 

1500 

3600 

4  23 

0  15 

5  47 

3  14 

0  25 

0  14 

1400 

3700 

4  17 

0  18 

5  39 

3  10 

0  29 

0  17 

1300 

3800 

4  11 

0  20 

5  30 

3  5 

0  33 

0  19 

1200 

3900 

4  4 

0  23 

5  21 

3  0 

0  37 

0  22 

1100 

4000 

3  57 

0  26 

5  12 

2  54 

0  41 

0  25 

1000 

4100 

3  49 

0  29 

5  2 

2  49 

0  45 

0  28 

900 

4200 

3  41 

0  32 

4  52 

2  43 

0  50 

0  31 

800 

4300 

3  33 

0  35 

4  41 

2  37 

0  54 

0  35 

700 

4400 

3  24 

0  39 

4  30 

2  30 

0  59 

0  38 

600 

4500 

3  15 

0  42 

4  19 

2  24 

1  4 

0  42 

500 

4600 

3  7 

0  46 

4  7 

2  17 

1  9 

0  45 

400 

4700 

2  58 

0  49 

3  56 

2  10 

1  14 

0  49 

3  00 

4800 

2  48 

0  53 

3  44 

2  4 

1  19 

0  53 

200 

4900 

2  3? 

0  56 

3  32 

1  57 

1  25 

0  56 

100 

5000 

2  30 

1  0 

3  20 

1  50 

1  30 

1  0 

0000 

5100 

2  21 

1  4 

3  8 

1  43 

1  35 

1  4 

9900 

5200 

2  11 

1  7 

2  56 

1  36 

1  40 

1  7 

9800 

5300 

2  2 

1  11 

2  44 

1  29 

1  46 

1  11 

9700 

5400 

1  53 

1  14 

2  33 

1  23 

1  51 

1  15 

9600 

5500 

1  44 

1  18 

2  21 

1  16 

1  56 

1  18 

9500 

5600 

1  36 

1  21 

2  10 

1  10 

2  1 

1  22 

9400 

5700 

1  27 

1  25 

1  59 

1  3 

2  6 

1  25 

9300 

5800 

1  19 

1  28 

1  48 

0  57 

2  10 

1  28 

9200 

5900 

1  11 

1  31 

1  38 

0  51 

2  15 

1  32 

9100 

6000 

1  3 

1  34 

1  28 

0  46 

2  19 

1  35 

9000 

6100 

0  56 

1  37 

1  19 

0  40 

2  23 

1  38 

8900 

6200 

0  49 

1  39 

1  10 

0  35 

2  27 

1  40 

8800 

6300 

0  43 

1  42 

1  1 

0  30 

2  31 

1  43 

8700 

6400 

0  36 

1  44 

0  53 

0  26 

2  35 

1  46 

8600 

6500 

0  31 

1  47 

0  46 

0  21 

2  38 

1  48 

8500 

6600 

0  26 

1  49 

0  39 

0  18 

2  41 

1  50 

8400 

6700 

0  21 

1  51 

0  33 

0  14 

2  43 

1  52 

8300 

6800 

0  17 

1  52 

0  28 

0  11 

2  46 

1  54 

8200 

6900 

0  13 

1  54 

0  23 

0  8 

2  48 

1  55 

8100 

7000 

0  10 

1  55 

0  19 

0  6 

2  50 

1  56 

8000 

7100 

0  7 

1  56 

0  16 

0  4 

2  51 

1  57 

7900 

7200 

0  5 

1  57 

0  13 

0  2 

2  52 

1  58 

7800 

7300 

0  4 

1  57 

0  11 

0  1 

2  53 

1  59 

7700 

7400 

0  3 

1  58 

0  10 

0  1 

2  54 

1  59 

7600 

7500 

0  3 

1  58 

0  10 

0  1 

2  54 

1  59 

7500 

40  TABLE  XXVII. .  TABLE  XXVIII. 


Equations  8  and  9.  Equations  10  and  11. 


Arg. 

8 

9 

Arg. 

8 

9 

Arg 

10 

11 

Arg. 

10 

11 

0 

V  20" 

V  20// 

5000 

1'  20" 

1'  20" 

0 

10" 

10" 

500 

10" 

10" 

100 

1  15 

1  29 

5100 

1  24 

1  26 

10 

9 

11 

510 

10 

11 

200 

1  11 

1  37 

5200 

1  29 

1  31 

20 

9 

12 

520 

9 

11 

300 

1  7 

1  46 

5300 

1  33 

1  37 

30 

8 

13 

530 

9 

12 

400 

1  2 

1  54 

5400 

1  37 

1  42 

40 

7 

14 

540 

8 

13 

500 

0  58 

2  1 

5500 

1  42 

1  47 

50 

7 

15 

550 

8 

14 

600 

0  54 

2  8 

5600 

1  46 

1  51 

60 

6 

16 

560 

8 

14 

700 

0  50 

2  15 

5700 

1  50 

1  55 

70 

6 

17 

570 

8 

15 

800 

0  46 

2  20 

5800 

1  54 

1  58 

80 

5 

17 

580 

7 

15 

900 

0  42 

2  25 

5900 

1  58 

2  0 

90 

5 

18 

590 

7 

15 

1000 

0  38 

2  29 

6000 

2  1 

2  1 

100 

5 

18 

600 

7 

16 

1100 

0  35 

2  32 

6100 

2  5 

2  2 

110 

4 

19 

610 

7 

16 

1200 

0  31 

2  34 

6200 

2  8 

2  2 

120 

4 

19 

620 

7 

16 

1300 

0  28 

2  35 

6300 

2  11 

2  1 

130 

4 

19 

630 

7 

16 

1400 

0  25 

2  35 

6400 

2  14 

1  59 

140 

4 

19 

640 

7 

15 

1500 

0  23 

2  34 

6500 

2  17 

1  56 

150 

4 

19 

650 

8 

15 

1600 

0  20 

2  32 

6600 

2  19 

1  52 

160 

4 

19 

660 

8 

15 

1700 

0  18 

2  29 

6700 

2  22 

1  48 

170 

4 

18 

670 

8 

14 

1800 

0  16 

2  26 

6800 

2  24 

1  43 

180 

5 

18 

680 

9 

13 

1900 

0  14 

2  21 

6900 

2  25 

1  38 

190 

5 

17 

690 

9 

13 

2000 

0  13 

2  16 

7000 

2  27 

1  32 

200 

5 

16 

700 

10 

12 

2100 

0  11 

2  11 

7100 

2  28 

1  25 

210 

6 

16 

710 

10 

11 

2200 

0  10 

2  4 

7200 

2  29 

1  18 

220 

6 

15 

720 

11 

10 

2300 

0  10 

1  58 

7300 

2  30 

1  11 

230 

7 

14 

730 

11 

9 

2400 

: 

0  9 

1  51 

7400 

2  30 

1  4 

240 

7 

13 

740 

12 

9 

{2500 

0  9 

1  43 

7500 

2  31 

0  56 

250 

8 

12 

750 

12 

8 

;2600 

0  10 

1  36 

7600 

2  30 

0  49 

260 

8 

11 

760 

13 

7 

;2700 

0  10 

1  29 

7700 

2  30 

6  42 

270 

9 

10 

770 

13 

6 

2800 

0  11 

1  22 

7800 

2  29 

0  36  ! 

280 

9 

10 

780 

14 

5 

i.2900 

0  12 

1  15 

7900 

2  28 

0  29 

290 

10 

9 

790 

14 

4 

3000 

0  13 

1  8 

8000 

2  27 

0  24 

300 

10 

8 

800 

15 

o 

13100 

0  15 

1  2 

8100 

2  26 

0  18 

310 

11 

7 

810 

15 

3 

3200 

0  16 

0  57 

8200 

2  24 

0  14 

320 

11 

6 

820 

15  ; 

1  2 

3300 

0  18 

0  52 

8300 

2  22 

0  10 

330 

12 

6 

8301 

1  16  { 

l  2 

,3400 

0  21 

0  47 

8400 

2  20 

0  8 

340 

12 

5 

840 

16  i 

i  1 

!3500 

0  23 

0  44 

8500 

2  17 

0  6 

350 

12 

5 

850 

16 

1 

13600 

0  26 

0  41 

8600 

2  15 

0  5 

360 

12 

5 

860 

16 

1 

{3700 

0  29 

0  39 

8700 

2  12 

0  5 

370 

13 

4 

870 

16 

1 

3800 

0  32 

0  38 

8800 

2  9 

0  6 

380 

13 

4 

880 

16 

1 

3900 

0  35 

0  38 

8900 

2  5 

0  8 

390 

13 

4 

890 

16 

1 

4000 

0  39 

0  39 

9000 

2  2 

0  11 

400 

13 

4 

900 

15 

o 

4100 

0  42 

0  40 

9100 

1  58 

0  15 

410 

13 

5 

910 

15 

2 

4200 

0  46 

0  42 

9200 

1  54 

0  20 

420 

12 

5 

920 

15 

3 

4300 

0  50 

0  45 

9300 

1  50 

0  25 

430 

12 

5 

930 

14 

3 

4400 

0  54 

0  49 

9400 

1  46 

0  32 

•  • 

440 

12 

6 

940 

14 

4 

4500 

0  58 

0  53 

9500 

1  42 

0  39 

450 

12 

6 

950 

13 

1  5 

4600 

1  3 

0  58 

9600 

1  58 

0  46 

460 

11 

7 

960 

13 

6 

4700 

1  7 

1  3 

9700 

1  33 

0  54 

470 

11 

8 

970 

12 

7 

4800 

1  11 

1  9 

9800 

1  29 

1  3 

480 

11 

8 

980 

11 

8 

4900 

1  16 

1  14 

9900 

1  24 

1  11 

490 

10 

9 

990 

11 

9 

[5000 

1  20 

l  20 

10000 

1  20 

1  20 

500 

10 

10 

1000 

10 

10  1 

TABLE  XXIX, 


TABLE  XXX.  41 

Equation  20, 


Equations  12  to  19. 


Arg. 

12 

13 

14 

15 

16 

17 

18 

19 

Arg 

.  20 

|Arg. 

250 

2' 

2" 

8' 

0' 

34' 

3'1 

'  17' 

'  3' 

'  250 

0 

10' 

'  500 

260 

2 

2 

8 

0 

34 

3 

17 

3 

240 

10 

11 

510 

270 

2 

2 

8 

0 

34 

3 

17 

3 

230 

20 

12 

520 

280 

3 

2 

8 

0 

33 

3 

17 

3 

220 

30 

13 

530 

290 

3 

2 

8 

0 

33 

4 

16 

3 

210 

40 

13 

540 

300 

3 

2 

8 

0 

33 

4 

16 

3 

200 

50 

14 

550 

310 

o 

O 

3 

9 

1 

33 

4 

16 

o 

O 

190 

60 

15 

560 

320 

4 

3 

9 

1 

32 

4 

16 

4 

180 

70 

16 

570 

330 

4 

4 

9 

1 

32 

4 

16 

4 

170 

80 

16 

580 

340 

5 

4 

10 

2 

32 

4 

16 

4 

160 

90 

17 

590 

350 

6 

5 

10 

2 

31 

5 

15 

4 

150 

100 

17 

600 

360 

6 

6 

11 

2 

31 

5 

15 

5 

140 

110 

17 

610 

370 

7 

7 

11 

3 

30 

5 

15 

5 

130 

120 

17 

/  620 

380 

■  8 

7 

12 

3 

29 

5 

15 

5 

120 

130 

17 

630 

390 

9 

8 

12‘ 

4 

29 

6 

14 

6 

110 

140 

17 

640 

400 

10 

9 

13 

4 

28 

6 

14 

6 

100 

150 

17 

650 

410 

10 

10 

1.3 

5 

27 

6 

14 

6 

90 

160 

17 

660 

420 

11 

11 

14 

5 

27 

7 

13 

7 

80 

170 

16 

670 

430 

12 

12 

15 

6 

26- 

7 

13 

7 

70 

180 

16 

680 

440 

13 

13 

15 

6 

25 

8 

12 

7 

60 

190 

15 

690 

450 

14 

14 

16 

7 

24 

8 

12 

8 

50 

200 

14 

700 

460 

16 

15 

17 

7 

23 

8 

12 

8 

40 

210 

13 

710 

470 

17 

16 

18 

8 

23 

9 

11 

9 

30 

220 

13 

720 

480 

18 

18 

18 

9 

22 

9 

11 

9 

20 

230 

12 

730 

490 

19 

19 

19 

9 

21 

10 

10 

10 

10 

240 

11 

740 

500 

20 

20 

20 

10 

20 

10 

10 

10 

000 

250 

10 

750 

510 

21 

21 

21 

11 

19 

10 

10 

10 

990 

260 

9 

760 

520 

22 

22 

21 

11 

18 

11 

9 

11 

980 

270 

8 

770 

530 

23 

23 

22 

12 

17 

11 

9 

11 

970 

280 

7 

780 

540 

24 

25 

23 

12 

17 

12 

8 

12 

960 

290 

6 

790 

550 

25 

26 

24 

13 

16 

12 

8 

12 

950 

300 

6 

800 

560 

26 

27 

24 

14 

15 

12 

7 

13 

940 

310 

5 

810 

570 

27 

28 

25 

.  14- 

14 

13 

7 

13 

930 

320 

4 

820 

580 

28 

29 

26 

15 

13 

13 

7 

13 

920 

330 

4 

830 

590 

29 

30 

26 

15 

13 

13 

6 

14 

910 

340 

O 

840 

600 

30 

31 

27 

16 

12 

14 

6 

14 

900 

350 

3 

850 

610 

31 

32 

28 

16 

11 

14 

6 

14 

890 

360 

3 

860 

620 

32 

33 

28 

17 

11 

14 

5 

15 

880 

370 

3 

870 

630 

33 

33 

29 

17 

10 

15 

5 

15 

870 

380 

3 

880 

640 

34 

34 

29 

18 

9 

15 

5 

15 

860 

390 

3 

890 

650 

34 

35 

30 

18 

9 

15 

5 

16 

850 

400 

3 

900' 

660 

35 

36 

30 

18 

8 

16 

4 

16 

840 

410 

3 

910 

670 

35 

36 

31 

19 

8 

16 

4 

16 

830 

420 

4 

920 

680 

36 

37 

31 

19 

8 

16 

4 

16 

820 

430 

4 

930 

690 

36 

37 

31 

19 

7 

16 

4 

17 

810 

440 

5 

940 

700 

37 

37 

32 

19 

7 

16 

4 

17 

800 

450 

6 

950 

710 

37 

38 

32 

20 

7 

16 

3 

17 

790 

160 

6 

960 

720 

37 

38 

32 

20 

6 

16 

3 

17 

780 

170 

7 

970 

730 

38 

38 

32 

20 

6 

16 

3 

17 

770 

i 

180 

8 

980 

740 

38 

38 

32 

20 

6 

17 

3 

17 

760 

A 

190 

9 

990 

750 

38 

38 

32 

20 

6 

17 

3 

17 

750 

i 

>00 

10  ] 

L000 

6* 


42 


TABLE  XXXI 


Erection. 

Argument.  Evection,  corrected, 


0* 

Is 

1  Is 

lil 

S 

IV 

S 

1. 

V« 

1 

0° 

1°30 

0" 

2°  10' 43" 

2°  40' 10" 

2°  50' 25" 

2°  39'  8" 

2°  9' 

43" 

1 

1 

31 

25 

2 

11 

57 

2 

40 

51 

2 

50 

23 

2 

38 

25 

2 

8 

29 

2 

1 

32 

51 

2 

13 

9 

2 

41 

30 

2 

50 

20 

2  37 

40 

2 

7 

16 

3 

1 

34 

16 

2 

14 

21 

2  42 

8 

2 

50 

15 

2 

36 

55 

2 

6 

2 

4 

1 

35 

42 

2 

15 

31 

2 

42 

45 

2 

50 

9 

2 

36 

8 

2 

4 

47 

5 

1 

37 

7 

2 

16 

41 

2 

43 

21 

2 

50 

1 

2 

35 

19 

2 

3 

32 

6 

1 

38  32 

2 

17 

50 

2 

43 

55 

2 

49 

52 

2 

34 

30 

2 

2 

16 

7 

1 

39 

57 

2 

18 

58 

2 

44 

27 

2 

49 

41 

2 

33 

40 

2 

1 

0 

8 

1 

41 

21 

2 

20 

5 

2 

44 

59 

2 

49 

29 

2 

32 

48 

1 

59 

43 

9 

1 

42 

46 

2 

21 

11 

2 

45 

29 

2 

49 

15 

2 

31 

55 

1 

58 

26 

10 

1 

44 

10 

2 

22 

17 

2 

45 

57 

2 

49 

0 

2  31 

2 

1 

57 

8 

11 

1 

45 

34 

2 

23 

21 

2 

46 

24 

2 

48 

43 

2 

30 

7 

1 

55 

49 

12 

1 

46 

58 

2 

24 

24 

2 

46 

50 

2 

48 

26 

2 

29 

11 

1 

54 

30 

13 

1 

48 

21 

2 

25 

26 

2 

47 

14 

2 

48 

6 

2 

28 

14 

1 

53 

11 

14 

1 

49 

44 

2 

26 

28 

2 

47 

37 

2 

47 

45 

2 

27 

16 

1 

51 

51 

15 

1 

51 

7 

2 

27 

28 

2 

47 

59 

2 

47 

23 

2 

26 

17 

1 

50 

31 

16 

1 

52 

29 

2 

28 

27 

2 

48 

19 

2 

47 

0 

2 

25 

17 

1 

49 

11 

17 

1 

53 

51 

2 

29 

25 

2 

48 

37 

2  46 

35 

2 

24 

16 

1 

47 

50 

18 

1 

55 

12 

2 

30 

21 

2 

48 

54 

2 

46 

8 

2 

23 

14 

1 

46 

29 

19 

1 

56  33 

2 

31 

17 

2 

49 

10 

2 

45 

41 

2 

22 

11 

1 

45 

7 

20 

1 

57 

53 

2  32 

11 

2 

49 

24 

2 

45 

12 

2 

21 

7 

1 

43 

46 

21 

1 

59 

13 

2 

33 

5 

2 

49 

37 

2 

44 

41 

2 

20 

2 

1 

42  24 

22 

2 

0  32 

2 

33 

57 

2 

49 

48 

2 

44 

9 

2 

18 

56 

1 

41 

2 

23 

2 

1 

51 

2 

34 

48 

2 

49 

58 

2 

43 

36 

2 

17 

50 

1 

39 

39 

24 

2 

3 

9 

2 

35 

38 

2 

50 

6 

2 

43 

2 

2 

16 

43 

1 

38 

17 

25 

*2 

4 

26 

2 

36 

26 

2 

50 

13. 

2 

42 

26 

:  2 

15 

34 

1 

36 

54 

26 

2 

5 

43 

2 

37 

13 

2 

50 

19 

2 

41 

49 

2 

14 

25 

1 

35 

32 

27 

2 

6 

59 

2 

37 

59 

2 

50 

23 

2 

41 

11 

2 

13 

16 

1 

34 

9 

28 

2 

8 

15 

2 

38 

44 

2 

50 

25 

2 

40 

31 

2 

12 

5 

1 

32 

46 

29 

2 

9 

30 

2 

39 

28 

2 

50 

26 

2 

39 

50 

2 

10 

54 

1 

31 

23 

30 

2 

10 

43 

2 

40 

10 

2 

50 

25 

2  39 

8 

2 

9 

42 

1 

30 

0 

TABLE  XXXI, 


43 


Evection. 

Argument.  Evection,  corrected. 


Yl* 

VIIs 

VIII* 

IX* 

X* 

XI 

i 

0° 

1‘ 

30' 

0" 

0‘ 

>50' 

18" 

0° 

20'  52" 

0°  9' 34" 

0' 

>19' 50" 

0‘ 

>49' 16" 

1 

1 

28 

37 

0 

49 

6 

0 

20 

10 

0 

9 

34 

0 

20 

32 

0 

50 

30 

2 

1 

27 

14 

0 

47 

55 

0 

19 

29 

0 

9 

35 

0 

21 

16 

0 

51 

45 

3 

1 

25 

51 

0 

46 

44 

0 

18 

49 

0 

9 

37 

0 

22 

1 

0 

53 

1 

4 

1 

24 

28 

0 

45 

34 

0 

18 

11 

0 

9 

41 

0 

22 

47 

0 

54 

17 

5 

1 

23 

6 

0 

44 

26 

0 

17 

34 

0 

9 

47 

0 

23 

34 

0 

55 

33 

6 

1 

21 

43 

0 

43 

17 

0 

16 

58 

0 

9 

54 

0 

24 

22 

0 

56 

51 

7 

1 

20 

20 

0 

42 

10 

0 

16 

24 

0 

10 

2 

0 

25 

12 

0 

58 

9 

8 

1 

18 

58 

0 

41 

4 

0 

15 

50 

0 

10 

12 

0 

26 

3 

0 

59 

28 

9 

1 

17 

36 

0  39 

58 

0 

15 

19 

0 

10 

23 

0 

26 

55 

1 

0 

47 

10 

1 

16 

14 

0 

38 

53 

0 

14 

48 

0 

10 

36 

0 

27 

48 

1 

2 

7 

11 

1 

14 

52 

0  37  49 

0 

14 

19 

0 

10 

50 

0 

28 

43  ' 

1 

3 

27 

12 

1 

13 

31 

0 

36 

46 

0 

13 

51 

0 

11 

5 

0 

29 

39 

1 

4  48 

13 

1 

12 

10 

0 

35 

44 

0 

13 

25 

0 

11 

23 

0 

30 

35 

1 

6 

9 

14 

1 

10 

49 

0 

34  43 

0 

13 

0 

0 

11 

41 

0 

31 

33 

1 

7 

31 

15 

1 

9 

29 

0 

33 

43 

0 

12 

37 

0 

12 

1 

0 

32  32 

1 

8 

53 

16 

1 

8 

9 

0 

32 

44 

0 

12 

14 

0 

12 

23 

0 

33 

32 

1 

10 

16 

17 

1 

6 

49 

0 

31 

46 

0 

11 

54 

0 

12 

45 

0 

34 

34 

1 

11 

39 

18 

1 

5 

30 

0 

30 

49 

0 

11 

34 

0 

13 

10 

0 

35 

36 

1 

13 

2 

19 

1 

4 

11 

0 

29 

53 

0 

11 

16 

0 

13 

35 

0  36 

39 

1 

14 

26 

20 

1 

2 

52 

0 

28 

58 

0 

11 

0 

0 

14 

3 

0 

37  43 

1 

15 

50 

21 

1 

1 

34 

0 

28 

5 

0 

10 

45 

0 

14 

31 

0 

38 

48 

1 

17 

14 

22 

1 

0 

17 

0 

27 

12 

0 

10 

31 

0 

15 

1 

0 

39 

55 

1 

18 

39 

23 

0 

59 

0 

0 

26 

20 

0 

10 

19 

0 

15 

33 

0 

41 

2 

1 

20 

3 

24 

0 

57  44 

0 

25 

30 

0 

10 

8 

0 

16 

5 

0 

42 

10 

1 

21 

28 

25 

0 

56 

28 

0 

24  40 

0 

9 

5§ 

0 

16 

39 

0 

43 

19 

1 

22 

53 

26 

0 

55 

13 

0 

23 

52  i 

0 

9 

51 

0 

17 

15 

0  44 

29 

1 

24 

18 

27 

0 

53 

58 

0 

23 

5  ! 

0 

9 

45 

0 

17 

52 

0 

45 

39 

1 

25 

44 

28 

0 

52 

44 

0 

22 

20  | 

0 

9  40 

0 

18 

30 

0 

46 

51 

1 

27 

9 

29 

0 

51 

31 

0 

21 

35 

0 

9 

36 

0 

19 

9 

0 

48 

3 

1 

28  34 

30 

0 

50 

18 

0 

20 

52 

0 

9 

34 

0 

19 

50 

0 

49 

16 

1 

30 

0 

TABLE  XXXII 


44 


Equation  of  Moon's  Centre . 
Argument.  Anomaly,  corrected. 


0* 

1S 

11* 

ill* 

IV* 

V* 

0° 

7° 

O' 

0" 

10 

20' 

58" 

12°  38' 44" 

13°  17' 

35" 

12°  16' 

21'' 

9°  58' 

29" 

1 

7 

7 

5 

10 

26 

52 

12 

41 

43 

13 

17 

5 

12 

12 

48 

9 

52 

58 

2 

7 

14 

10 

10 

32 

42 

12 

44 

35 

13 

16 

28 

12 

9 

11 

9 

47 

24 

3 

7 

21 

15 

10 

38 

27 

12 

47 

20 

13 

15 

44 

12 

5 

29 

9 

41 

48 

4 

7 

28 

19 

10 

44 

8 

12 

49 

59 

13 

14 

53 

12 

1 

41 

9 

36 

10 

5 

7 

35 

23 

10 

49 

43 

12 

52 

30 

13 

13 

56 

11 

57 

49 

9 

30 

29  I 

6 

7 

42 

26 

10 

55 

14 

12 

54 

55 

13 

12 

52 

11 

53 

52 

9 

24 

46 

7 

7 

49 

28 

11 

0 

39 

12 

57 

12 

13 

11 

41 

11 

49 

50 

9 

19 

1 

8 

7 

56 

28 

11 

6 

0 

12 

59 

23 

13 

10 

24 

11 

45 

44 

9 

13 

13 

9 

8 

3 

28 

11 

11 

15 

13 

1 

26 

13 

9 

1 

11 

41 

33 

9 

7 

24 

10 

8 

10 

26 

11 

16 

24 

13 

3 

23 

13 

7 

31 

. 

11 

37 

17 

9 

1 

52  : 

11 

8 

17 

22 

11 

21 

29 

13 

5 

12 

13 

5 

54 

11 

32 

57 

8 

55 

39 

12 

8 

24 

17 

11 

26 

27 

13 

6 

55 

13 

4 

12 

11 

28 

33 

8 

49 

44 

13 

8 

31 

10 

11 

31 

20 

13 

8 

30 

13 

2 

23 

11 

24 

5 

8 

43 

47 

14 

8 

38 

1 

11 

36 

8 

13 

9 

59 

13 

0 

27 

11 

19 

32 

8 

37 

49 

15 

8 

44 

50 

11 

40 

49 

13 

11 

20 

12 

58 

26 

11 

14 

55 

8 

31 

49 

16 

8 

51 

36 

11 

45 

25 

13 

12 

34 

12 

56 

18 

11 

10 

14 

8 

25 

48 

17 

8 

58 

20 

11 

49 

54 

13 

13 

41 

12 

54 

5 

11 

5 

30 

8 

19 

46 

18 

9 

5 

1 

11 

54 

18 

13 

14 

41 

12 

51 

45 

11 

0 

41 

8 

13 

42' 

19 

9 

11 

39 

11 

58 

35 

13 

15 

34 

12 

49 

19 

10 

55 

49 

8 

7 

38 

20 

9 

18 

15 

12 

2 

47 

13 

16 

20 

12 

46 

47 

10 

50 

53 

8 

1 

32 

21 

9 

24 

47 

12 

6 

52 

13 

16 

59 

12 

44 

10 

10 

45 

53 

7 

55 

26 

22 

9 

31 

16 

i  12 

10 

50 

13 

17 

31 

12 

41 

27 

10 

40 

50 

7 

49 

18 

23 

9 

57 

42 

12 

14 

42 

13 

17 

56 

12 

38 

38 

10 

35 

43 

7 

43 

10 

24 

9 

44 

4 

!12 

18 

28 

13 

18 

14 

12 

35 

43 

10 

30 

33 

7 

37 

1 

25 

9 

50 

23 

12 

22 

7 

13 

18 

24 

12 

32 

43 

10 

25 

20 

7 

30 

52 

26 

9 

56 

38 

12 

25 

40 

13 

18 

28 

12 

29 

37 

10 

20 

4 

7 

24 

42 

27 

10 

2 

49 

12 

29 

6 

13 

18 

25 

12 

26 

26 

10 

14 

45 

7 

18 

32 

28 

10 

8 

56 

12 

32 

25 

13 

18 

16 

12 

23 

10 

10 

9 

22 

7 

12 

21 

29 

10 

14 

59 

12 

35 

38 

13 

17 

59 

12 

19 

48 

10 

3 

57 

7 

6 

11 

30 

10 

20 

58 

;12 

38 

44 

13 

17 

35 

12 

16 

21 

9 

58 

29 

7 

0 

0 

TABLE  XXXII, 


45 


Equat  ion  of  Moon's  Centre . 
Argument.  Anomaly,  corrected. 


--ft— — 

VI* 

VII* 

VIIIs 

IX* 

Xs 

XI* 

0° 

7°  O'  0" 

4°  1'  31" 

1°43'39" 

0°42'25" 

i°2r’i6" 

3°  39'  2" 

1 

6  53  49 

3  56  3 

1  40  12 

0  42  1 

1  24  22 

3  45  1 

2 

6  47  39 

3  50  38 

1  36  50 

0  41  44 

1  27  35 

3  51  4 

3 

6  41  28 

3  45  15 

1  33  34 

0  41  35 

1  30  54 

3  57  11 

4 

6  35  18 

3  39  56 

1  30  23 

0  41  32 

1  34  20 

4  3  22 

5 

6  29  8 

3  34  40 

1  27  17 

0  41  36 

1  37  53 

4  9  37 

6 

6  22  59 

3  29  26 

1  24  17 

0  41  46 

1  41  32 

4  15  55 

r 

6  16  50 

3  24  17 

1  21  22 

0  42  4 

1  45  18 

4  22  18 

8 

6  10  42 

3  19  10 

1  18  33 

0  42  29 

1  49  10 

4  28  44 

9 

6  4  34 

3  14  7 

1  15  50 

0  43  1 

153  8 

4  35  13 

10 

5  58  28 

3  9  7 

1  13  12 

0  43  40 

1  57  13 

4  41  45 

11 

5  52  22 

3  4  11 

1  10  41 

0  44  26 

2  1  24 

4  48  21 

12 

5  46  17 

2  59  19 

1  8  15 

0  45  19 

2  5  42 

4  54  59 

13 

5  40  14 

2  54  30 

1  5  55 

0  46  19 

2  10  5 

5  1  40 

14 

5  34  12 

2  49  46 

1  3  42 

0  47  26 

2  14  35 

5  8  24 

15 

5  28  11 

2  45  5 

1  1  34 

0  48  40 

2  19  11 

5  15  10 

16 

5  22  11 

2  40  28 

0  59  33 

0  50  1 

2  23  52 

5  21  59 

17 

5  16  13 

2  35  55 

0  57  37 

0  51  30 

2  28  39 

5  28  50 

18 

5  10  16 

2  31  27 

0  55  48 

0  53  5 

2  33  32 

5  35  43 

19 

5  4  21 

2  27  3 

0  54  6 

0  54  47 

2  38  31 

5  42  37 

20 

4  58  28 

2  22  43 

0  52  29 

0  56  37 

2  43  35 

5  49  34 

21 

4  52  36 

2  18  27 

0  50  59 

0  58  33 

2  48  45 

5  56  32 

22 

4  46  47 

2  14  16 

0  49  36 

1  0  37 

2  54  0 

6  3  31 

23 

4  40  59 

2  10  10 

0  48  19 

1  2  48 

2  59  21 

6  10  32 

24 

4  35  14 

2  6  8 

0  47  8 

15  5 

3  4  46 

6  17  34 

25 

4  29  31 

2  2  11 

0  46  4 

1  7  30 

3  10  17 

6  24  37 

26 

4  23  50 

1  58  19 

0  45  7 

1  10  1 

3  15  52 

6  31  41 

27 

4  18  11 

1  54  31 

0  44  16 

1  12  40 

3  21  33 

6  38  45 

28 

4  12  35 

1  50  49 

0  43  32 

1  15  25 

3  27  18 

6  45  50 

29 

4  7  2 

1  47  11 

0  42  55 

1  18  17 

3  33  8 

6  52  55 

30 

4  1  31 

1  43  39 

0  42  25 

1  21  16 

3  39  2 

7  0  0 

46 


TABLE  XXXIII 


\ 


Variation * 

Argument.  Variation,  corrected. 


0* 

Is 

11* 

111* 

IV1 

i 

V* 

0° 

0°38/ 

0" 

lc 

8' 

1" 

1°  6- 

58" 

0°35/54" 

0° 

'  5'  29" 

0‘ 

3  6' 

2" 

1 

0 

39 

13 

1 

8 

35 

1 

6 

18 

0 

34 

40 

0 

4 

54 

0 

6 

42 

2 

0 

40 

26 

1 

9 

7 

1 

5 

36 

0 

33 

27 

0 

4 

21 

0 

7 

24 

3 

0 

41 

39 

1 

9 

36 

1 

4 

52 

0 

32 

13 

0 

3 

51 

0 

8 

8 

4 

0 

42 

52 

1 

10 

3 

1 

4 

5 

0 

31 

0 

0 

3 

22 

0 

8 

55 

5 

0  44 

4 

1 

10 

28 

1 

3 

17 

0 

29 

47 

0 

2 

56 

0 

9 

44 

6 

0 

45 

16 

1 

10 

50 

1 

2 

27 

0 

28 

34 

0 

2  33 

0 

10 

34 

7 

0  46 

28 

1 

11 

9 

1 

1 

35 

0 

27 

22 

0 

2 

12 

0 

11 

27 

8 

0  47  38 

1 

11 

26 

1 

0 

42 

0 

26 

11 

0 

1 

54 

0 

12 

22 

9 

0 

48 

48 

1 

11 

41 

0 

59  4-6 

0 

25 

1 

0 

1 

38 

0 

13 

19  ’ 

10 

0 

49 

57 

1 

11 

53 

0 

58 

49 

0 

23 

51 

0 

1 

24 

0 

14 

17 

11 

0 

51 

6 

1 

12 

2 

0 

57 

50 

0 

22 

42 

0 

1 

14 

0 

15 

17 

12 

0 

52 

13 

1 

12 

9 

0 

56 

50 

0 

21 

34 

0 

1 

5 

0 

16 

19 

13 

0 

53 

19 

1 

12 

13 

0 

55 

48 

0 

20 

28 

0 

1 

0 

0 

17 

22 

14 

0 

54 

24 

1 

12 

15 

0 

54 

45 

0 

19 

22 

0 

0 

57 

0 

18 

27 

15 

0 

55 

27 

1 

12 

14 

0 

53 

41 

0 

18 

18 

0 

0 

57 

0 

19 

33 

16 

0 

56 

30 

1 

12 

10 

0 

52 

35 

0 

17 

15 

0 

0 

59 

0 

20  41 

17 

0 

57 

31 

1 

12 

4 

0 

51 

28 

0 

16 

13 

0 

1 

4 

0 

21 

50 

18 

0 

58 

30 

1 

11 

55 

0 

50 

21 

0 

15 

13 

0 

1 

11 

0 

23 

0 

19 

0 

59 

28 

1 

11 

44 

0 

49 

12 

0 

14 

15 

0 

1 

22 

0 

24 

11 

20 

1 

0 

24 

1 

11 

30 

0  48 

2 

0 

13 

17 

!  0 

1 

34 

0 

25 

23 

21 

1 

1 

19 

1 

11 

14  ! 

!  o 

46 

52 

0 

12 

22 

0 

1 

50 

0 

26  36 

22 

1 

2 

11 

1 

10 

55 

0 

45 

40 

0 

11 

28 

0, 

2 

8 

0 

27 

50 

23 

1 

3 

2 

1 

10 

34 

0 

44 

29 

0 

10 

37 

0 

2 

28 

0 

29 

4 

24 

1 

3 

51 

1 

10 

10 

0 

43 

16 

0 

9 

47 

0 

2 

51 

0  30 

20 

25 

1 

4 

38 

1 

9 

44 

0 

42 

3 

0 

8 

59 

0 

3 

17 

0 

31 

36 

26 

1 

5 

23  ' 

i  1 

9 

15 

0  40 

50 

0 

8 

13 

0 

3 

45 

0 

32 

52 

.27 

1 

6 

6  1 

i  1 

8 

44 

0 

39 

36 

0 

7 

29 

0 

4 

16 

0 

34 

9 

28 

1 

6 

47  ! 

!  i 

8 

11 

0 

38 

22 

0 

6 

47 

0 

4 

48 

0 

35 

26 

29 

1 

7 

25  1 

1 1 

7 

36 

0 

37 

8 

0 

6 

7 

0 

5 

24 

0 

36 

43 

30 

1 

8 

1 

;  l 

6 

58 

0 

35 

54 

0 

5 

29 

0 

6 

2 

0 

38 

0 

TABLE  XXXIII, 


47 


Variation . 

Argument.  Variation,  corrected. 


VI* 

VIIs 

VIIIs 

IX* 

Xs 

XI* 

0° 

0°38 

0" 

lc 

9'  58" 

1°  10' 30" 

0°  40'  6" 

oc 

1  9' 

’  2" 

0°  7' 58" 

1 

0 

39 

17 

1 

10 

36 

1 

9 

53 

0  38 

52 

0 

8 

24 

0 

8  35 

2 

0 

40 

34 

1 

11 

11 

1 

9 

13 

0 

37 

38 

0 

7 

49 

0 

9  13 

3 

0 

41 

51 

1 

11 

44 

1 

8 

31 

0 

36 

24 

0 

7 

15 

0 

9  54 

4 

0 

43 

8 

1 

12 

15 

1 

7 

47 

0 

35 

10 

0 

6 

45 

0 

10  37 

5 

0 

44  24 

1 

12 

43 

1 

7 

1 

0 

33 

57 

0 

6 

16 

0 

11  22 

6 

0 

45 

40 

1 

13 

9 

1 

6 

13 

0 

32 

44 

0 

5 

50 

0 

12  9 

7 

0 

46 

55 

1 

13 

32 

1 

5 

23 

0 

31 

31 

0 

5 

26 

0 

12  58 

8 

0 

48 

10 

1 

13 

52 

1 

4 

31 

0 

30 

19 

0 

5 

5 

0 

13  49 

9 

0 

49 

24 

1 

14 

10 

1 

3 

38 

0 

29 

8 

0 

4 

46 

0 

14  41 

10 

0 

50 

37 

1 

14 

26 

1 

2 

42 

0 

27  58 

0 

4 

29 

0 

15  36 

11 

0 

51 

49 

1 

14 

38 

1 

1 

45 

0 

26 

48 

0 

4 

16 

0 

16  32 

12 

0 

53 

0 

1 

14 

48 

1 

0 

47 

0 

25  39 

0 

4 

4 

0 

17  30 

13 

0 

54 

10 

1 

14 

56 

0 

59 

47 

0 

24 

31 

0 

3 

56 

0 

18  29 

14 

0 

55 

19 

1 

15 

1 

0 

58 

45 

0 

23 

25 

0 

3 

50 

0 

19  30 

15 

0 

56 

27 

1 

15 

3 

0 

57 

42 

0 

22 

19 

0 

3 

46 

0 

20  32 

16 

0 

57 

o  o 
1)0 

1 

15 

3 

0 

56 

38 

0 

21 

15 

0 

3 

45 

0 

21  36 

17 

0 

58 

38 

1 

15 

0 

0 

55 

32 

0 

20 

12 

0 

3 

47 

0 

22  41 

18 

0 

59 

41 

1 

14 

54 

0 

54 

25 

0 

19 

10 

0 

3 

51 

0 

23  47 

19 

1 

0 

43 

1 

14 

46 

0 

53 

18 

0 

18 

10 

0 

3 

58 

0 

24  54 

20 

1 

1 

43 

1 

14 

35 

0 

52 

9 

0 

17 

11 

0 

4 

7 

0 

26  3 

21 

1 

2 

41 

1 

14 

22 

0 

50 

59 

0 

16 

14 

0 

4 

19 

0 

27  12 

22 

1 

3 

38 

1 

14 

6 

0  ■ 

49 

49 

0 

15 

18 

0 

4 

34 

0 

28  22 

23 

1 

4 

33 

1 

13 

48 

0  . 

48 

38 

0 

14 

25 

0 

4 

51 

0 

29  32 

24 

1 

5 

25 

1 

13 

27 

0 

47 

26 

0 

13 

33 

0 

5 

10 

0  30  44 

25 

1 

6 

16 

1 

13 

3 

0 

46 

13 

0 

12 

43 

0 

5 

32 

0 

31  55 

26 

1 

7 

5 

1 

12 

38 

0 

45 

0 

0 

11 

54 

0 

5 

57 

0 

33  8 

27 

1 

7 

52 

1 

12 

9 

0 

43 

47 

0 

11 

8 

0 

6 

23 

0 

34  20 

28 

1 

8 

36 

1 

11 

39 

0 

42 

33 

0 

10  24 

0 

6 

53 

0  35  33 

29 

1 

9 

18 

1 

11 

6 

0 

41 

20 

0 

9 

42 

0 

7 

24 

0 

36  47 

30 

1 

9 

58 

1 

10 

30 

0 

40 

6 

0 

9 

2 

0 

7 

58 

0 

38  0 

48 


TABLE  XXXIV, 


Reduction. 

Argument.  Suppl.  of  Node  -f  Moon’s  Orbit  Longitude. 


G* 

VIs 

1*  VIP 

IL*  VIII* 

ill*  IX* 

IV*  X* 

V* 

XI* 

0° 

7 

0' 

1'  3" 

V  3" 

•  7,  o// 

’  12'  57" 

'  12  '57" 

1 

6 

46 

0  56 

1  10 

7  14 

13  4 

12 

50 

2 

6 

31 

0  49 

1  18 

7  29 

13  10 

12 

42 

3 

6 

17 

0  43 

1  26 

7  43 

13  17 

12 

33 

4 

6 

3 

0  38 

1  35 

7  57 

13  22 

12 

25 

5 

5 

48 

0  33 

1  44 

8  12 

13  27 

12 

16 

6 

5 

34 

0  28 

1  54 

8  26 

13  32 

12 

6 

7 

5 

20 

0  24 

2  3 

8  40 

13  36 

11 

56 

8 

5 

6 

0  20 

2  14 

8  54 

13  40 

11 

46 

9 

4 

53 

0  17 

2  24 

9  7 

13  43 

11 

36 

10 

4 

39 

0  14 

2  35 

9  21 

13  46 

11 

25 

11 

4 

26 

0  12 

2  46 

9  34 

13  48 

11 

14 

12 

4 

12 

0  10 

2  58 

9  48 

13  50 

11 

2 

13 

3 

59 

0  9 

3  9 

10  1 

13  51 

10' 

50 

14 

3 

46 

0  8 

3  22 

10  13 

13  52 

10  38 

15 

3 

34 

0  8 

3  34 

10  26 

13  52 

10 

26 

16 

3 

22 

0  8 

3  46 

10  38 

13  52 

10 

13 

17 

3 

9 

0  9 

3  59 

10  50 

13  51 

10 

1 

18 

2 

58 

0  10 

4  12 

11  2 

13  50 

9 

48 

19 

2 

46 

0  12 

4  26 

11  14 

13  48 

9 

34 

20 

2 

35 

0  14 

4  39 

11  25 

13  46 

9 

21 

21 

2 

24 

0  17 

4  53 

11  36 

13  43 

9 

7 

22 

2 

14 

0  20 

5  6 

11  46 

13  40 

8 

54 

23 

2 

3 

0  24 

5  20 

11  56 

13  36 

8 

40 

24 

1 

54 

0  28 

5  34 

12  6 

13  32 

8 

26 

25 

1 

44 

0  33 

5  48 

12  16 

13  27 

8 

12 

26 

1 

35 

0  38 

6  3 

12  25 

13  22 

7 

57 

27 

1 

26 

0  43 

6  17 

12  33 

13  17 

7 

43 

28 

1 

18 

0  49 

6  31 

12  42 

13  10 

7  29 

29 

1 

10 

0  56 

6  46 

12  50 

13  4 

7 

14 

30 

1 

3 

1  3 

7  0 

12  57  | 

12  57 

7 

0 

TABLE  XXXV, 


49 


Moon’s  Distance  from  the  North  Pole  of  the  Ecliptic. 
Argument.  Suppl.  of  Node  -j-  Moon’s  Orbit  Longitude. 


III* 

IV* 

v,  , 

j  VI* 

Ml 

S 

Vlli* 

0° 

84° 39' 16" 

85°  20' 43" 

87° 13' 47" 

89°48/ 

'  0" 

92°  22' 

'  13" 

94°  15' 

17" 

30° 

1 

84 

39 

19 

85 

23 

27 

87 

18 

28 

89 

53 

23 

92 

26 

52 

94 

17 

57 

29 

2 

84 

39 

27 

85 

26 

16 

87 

23 

12 

89 

58 

46 

92 

31 

27 

94 

20 

31 

28 

3 

84 

39 

41 

85 

29 

10 

87 

27 

58 

90 

4 

8 

92 

36 

0 

94 

23 

1 

27 

4 

84 

40 

1 

85 

32 

9 

87 

32 

48 

90 

9 

31 

92 

40 

30 

94 

25 

25 

26 

5 

84 

40 

27 

85 

35 

12 

87 

37 

39 

90 

14 

52 

92 

44 

56 

94 

27 

45 

25 

6 

84 

40 

58 

85 

38 

20 

87 

42 

33 

90 

20 

14 

92 

49 

19 

94 

29 

59 

24 

7 

84 

41 

34 

85 

41 

33 

87 

47 

30 

90 

25 

35 

92 

53 

39 

94 

32 

8 

23 

8 

84 

42 

17 

85 

44 

50 

87 

52 

28 

90 

30 

55 

92 

57 

56 

94 

34 

12 

22 

9 

84 

43 

5 

85 

48 

11 

87 

57 

29 

90 

36 

14 

93 

2 

9 

94 

36 

11 

21 

10 

84 

43 

58 

85 

51 

37 

88 

2 

31 

90 

41 

33 

93 

6 

18 

94 

38 

4 

20 

11 

84 

44 

57 

85 

55 

7 

88 

7 

36 

90 

46 

50 

93 

10 

24 

94 

39 

52 

19 

12 

84 

46 

2 

85 

58 

42 

88 

12 

42 

90 

52 

7 

93 

14 

27 

94 

41 

35 

18 

13 

84 

47 

12 

86 

2 

20 

88 

17 

50 

90 

57 

22 

93 

18 

25 

94 

43 

13 

17 

14 

84 

48 

27 

86 

6 

3 

88 

23 

0 

91 

2 

36 

93 

22 

20 

94 

44 

45 

16 

15 

84 

49 

49 

86 

9 

50 

88 

28 

11 

91 

7 

49 

93 

26 

10 

94 

46 

11 

15 

16 

84 

51 

15 

86 

13 

40 

88 

33 

24 

91 

13 

0 

93 

29 

57 

94 

47 

32 

14 

17 

84 

52 

47 

86 

17 

35 

88 

38 

38 

91 

18 

10 

93 

33 

40 

94 

48 

48 

13 

18 

84 

54 

25 

86 

21 

33 

88 

43 

53 

91 

23 

18 

93 

37 

18 

94 

49 

58 

12 

19 

84 

56 

7 

86 

25 

36 

88 

49 

10 

91 

28 

24 

93 

40 

53 

94 

51 

3 

11 

20 

84 

57 

56 

86 

29 

42 

88 

54 

27 

91 

33 

29 

93 

44 

23 

94 

52 

2 

10 

21 

84 

59 

49  i 

86 

33 

51 

88 

59 

46 

91 

38 

31 

93 

47 

49 

94 

52 

55 

9 

22 

85 

1 

48 

|86 

38 

4 

89 

5 

5 

91 

43 

32 

93 

51 

10 

94 

53 

43 

8 

23 

85 

o 

O 

52 

86 

42 

21 

89 

10 

25 

91 

48 

30 

93 

54 

27 

94 

54 

26 

7 

24 

85 

6 

1 

86 

46 

41 

89 

15 

46 

I91 

53 

27 

93 

57 

40 

94 

55 

2 

6 

25 

85 

8 

15 

86 

51 

4 

89 

21 

7 

91 

58 

21 

94 

0 

48 

94 

55 

33 

5 

26 

85 

10 

35 

86 

55 

30 

89 

26 

29 

92 

3 

12 

94 

3 

51 

94 

55 

59 

4 

27 

85 

12 

59 

87 

0 

0 

89 

31 

52 

92 

8 

1 

94 

6 

50 

94 

56 

18 

O 

O 

28 

85 

15 

29 

87 

4 

32 

89 

37 

14 

92 

12 

48 

94 

9 

44 

94 

56 

33 

2 

29 

85 

18 

3 

87 

9 

8 

89 

42 

37 

92 

17 

32 

94 

12 

33 

94 

56 

41 

1 

30 

85 

20 

43 

87 

i 

13 

47 

89 

48 

0 

92 

22 

13 

94 

15 

17 

94 

56 

44 

0 

IP 

1* 

0* 

XI 

l 

X* 

IX* 

30 


TABLE  XXXVI 


Equation  II.  of  the  Moon's  Polar  Distance . 
Argument  II,  corrected. 


111> 

IV* 

V* 

VI* 

Vil* 

VIII* 

0° 

O'  14" 

1'  24" 

4'  3  7V 

9' 

0" 

13' 

23" 

16'  36" 

30° 

1 

0 

14 

1 

29 

4 

45 

9 

9 

13 

31 

16 

40 

29 

2 

0 

14 

1 

34 

4 

53 

9 

18 

13 

39 

16 

45 

28 

3 

0 

14 

1 

39 

5 

1 

9 

27 

13 

47 

16 

49 

27 

4 

0 

15 

1 

44 

5 

9 

9 

37 

13 

54 

16 

53 

26 

5 

0 

16 

1 

49 

5 

18 

9 

46 

14 

2 

16 

57 

25 

6 

0 

17 

1 

54 

5 

26 

9 

55 

14 

9 

17 

1 

24 

7 

0 

18 

2 

0 

5 

34 

10 

4 

14 

17 

17 

4 

23 

8 

0 

19 

O 

<6 

5 

5 

43 

10 

13 

14 

24 

17 

8 

22 

9 

0 

20 

2 

11 

5 

51 

10 

22 

14 

31 

17 

11 

21 

10 

0 

22 

2 

17 

6 

0 

10 

31 

14 

38 

17 

14 

20 

11 

0 

23 

2 

23 

6 

9 

10 

40 

14 

45 

17 

17 

19 

12 

0 

25 

2 

29 

6 

17 

10  49 

14 

52 

17 

20 

18 

13 

0 

27 

2 

35 

6 

26 

10 

58 

14 

59 

17 

23 

17 

14 

0 

29 

2 

41 

6 

35 

11 

7 

15 

5 

17 

26 

16 

15 

0 

32 

2 

48 

6 

44 

11 

16 

15 

12 

17 

28 

15 

16 

0 

34 

2 

54 

6 

53 

11 

25 

15 

18 

17 

31 

14 

17 

0 

37 

3 

1 

7 

2 

11 

34 

15 

25 

17  33 

13 

18 

0 

40 

3 

8 

7 

11 

11 

43 

15 

31 

17 

35 

12 

19 

0 

42 

3 

15 

7 

20 

11 

51 

15 

37 

17 

36 

11 

20 

0 

45 

3 

22 

7 

29 

12 

0 

15 

43 

17 

38 

10 

21 

0 

49 

3 

29 

7  38 

12 

9 

,  15 

49 

17 

40 

9 

22 

0 

52 

3 

36 

7 

47 

12 

17 

!  15 

55 

17 

41 

8 

23 

0 

56 

3 

43 

7 

56 

12 

26 

16 

0 

17 

42 

7 

24 

0 

59 

3 

51 

8 

5 

12 

34 

16 

6 

17 

43 

6 

25 

1 

3 

3 

58 

8 

14 

12 

42 

16 

11 

17 

44 

5 

26 

1 

7 

4 

6  , 

8 

23 

12 

51 

16 

16 

17 

45 

4 

27 

1 

11 

4 

13 

8 

32 

12 

59 

16 

21 

17 

45 

3 

28 

1 

15 

4 

21 

8 

42 

13 

7 

16 

26 

17 

46 

2 

29 

1 

20 

4 

29 

8 

51 

13 

15 

16 

31 

17 

46 

1 

30 

1 

24 

4  37 

9 

0 

13 

23 

16 

36 

17 

46 

0 

II* 

Is 

0s 

XI* 

X* 

IX* 

TABLE  XXXVII. 

Equation  III.  of  the  Polar  Distance. 


Argument  Moon’s  True  Longitude. 


111* 

IV* 

V* 

VI* 

VII* 

VIII* 

0° 

16" 

15" 

12" 

8" 

4" 

1" 

30° 

6 

16 

14 

11 

7 

3 

1 

24 

12 

16 

14 

10 

6 

3 

0 

18 

18 

16 

13 

10 

5 

2 

0 

12 

24 

15 

13 

9 

5 

1 

0 

6 

30 

15 

12 

8 

4 

1 

0 

0 

II* 

I* 

0* 

XI* 

X* 

IX* 

TABLE  XXXVIII 


TABLE  XXXIX 


51 

To  Convert  Degrees  and  Equations  of  Polar  Distance. 

Minutes  into  Decimal  Arguments.  20  of  Long. ;  V  to  IX,  cor- 
Parts.  rected;  and  X,  not  corrected. 


Degrees 

Dec. 

Arg. , 

20 

V. 

VI. 

VII. 

VIII 

IX.  : 

X. 

Arg.j 

ami  Min. 

parts. 

250 

0" 

56" 

6" 

3" 

25" 

3" 

Tr 

250 

1°  5' 

005 

260 

0 

56 

6 

3 

25 

3 

n 

240 

1  26 

4 

270 

0 

56 

6 

3 

25 

o 

O 

n 

230 

1  48 

5 

280 

1 

55 

6 

3 

25 

3 

n 

220 

2  10 

6 

290 

1 

55 

7 

3 

25 

4 

n 

210 

2  31 

7 

300 

1 

55 

7 

4 

25 

4 

TT 

200 

2  53 

8 

310 

1 

54 

8 

4 

24 

5 

12 

190 

3  14 

9 

320 

2 

53 

8 

5 

24 

6 

12 

180 

3  36 

10 

330 

2 

53 

9 

5 

24 

6 

13 

170 

3  58 

11 

340 

3 

52 

10 

6 

23 

7 

13 

160 

4  19 

12 

350 

3 

51 

11 

7 

23 

8 

14 

150 

4  41 

13 

360 

4 

50 

12 

8 

23 

9 

14 

140 

5  2 

14 

370 

4 

49 

13 

9 

22 

10 

15 

130 

5  24 

15 

380 

5 

48 

14 

10 

22 

11 

16 

120 

5  46 

16 

390 

6 

46 

15 

11 

21 

13 

17 

110 

6  7 

17 

400 

6 

45 

16 

12 

21 

14 

17 

100 

6  29 

18 

410 

7 

44 

17 

13 

20 

15 

18 

90 

6  50 

19 

420 

8 

42 

18 

14 

20 

17 

19 

80 

7  12 

20 

430 

9 

41 

20 

15 

19 

18 

20 

70 

7  34 

21 

440 

10 

39 

21 

17 

19 

20 

21 

60 

7  55 

22 

450 

10 

38 

23 

18 

18 

22 

22 

50 

8  17 

23 

460 

11 

36 

24 

19 

17 

23 

23 

40 

8  38 

24 

470 

12 

35 

25 

21 

17 

25 

24 

30 

9  0 

25 

480 

13 

33 

27 

22 

16 

27 

25 

20 

9  22 

26 

490 

14 

32 

28 

24 

16 

28 

26 

10 

9  43 

27 

500 

15 

30 

30 

25 

15 

30 

27 

000 

10  5 

28 

510 

16 

28 

31 

26 

14 

32 

28 

990 

10  26 

29 

520 

17 

27 

33 

28 

14 

33 

29 

980 

10  48 

30 

530 

18 

25 

34 

29 

13 

35 

30 

970 

11  10 

31 

540 

19 

24 

36 

31 

12 

37 

31 

960 

11  31 

32 

550 

19 

22 

37 

32 

12 

38 

32 

950 

11  53 

33 

560 

20 

20 

39 

33 

11 

40 

33 

940 

12  14 

34 

570 

21 

19 

40 

34 

11 

41 

34 

930 

12  36 

35 

580 

22 

17 

41 

36 

10 

43 

35 

920 

12  58 

36 

590 

23 

16 

43 

37 

10 

44 

36 

910 

13  19 

37 

600 

24 

15 

44 

38 

9 

46 

37 

900 

13  41 

38 

610 

24 

13 

45 

39 

9 

47 

37 

890 

14  2 

39 

620 

25 

12 

46 

40 

8 

48 

38 

880 

14  24 

40 

630 

26 

11 

47 

41 

8 

50 

39 

870 

14  46 

41 

640 

26 

10 

48 

42 

7 

51 

40 

860 

15  7 

42 

650 

27 

9 

~49~ 

43 

7 

52 

40 

850 

15  29 

43 

660 

27 

8 

50 

44 

6 

53 

41 

840 

15  50 

44 

670 

28 

7 

51 

45 

6 

54 

41 

830 

16  12 

45 

680 

28 

7 

52 

45 

6 

54 

42 

820 

16  34 

46 

690 

29 

6 

52 

46 

6 

55 

42 

810 

16  55 

47 

700 

29 

5 

53 

46 

5 

56 

42 

800 

17  17 

48 

710 

29 

5 

53 

47 

5 

56 

43 

790 

17  38 

49 

720 

29 

5 

53 

47 

5 

56 

43 

780 

18  0 

50 

730 

30 

4 

54 

47 

5 

57 

43 

770 

18  22 

51 

740 

30 

4 

54 

47 

5 

57 

43 

760 

18  43 

19  5 

52 

53 

750 

i  30 

4 

54 

47 

5 

57 

43 

•750 

TABLE  XL 


Morris  Equatorial  Parallax . 

Argument.  Argument  of  the  Evection. 


1 

1* 

11* 

III* 

1  n‘ 

V* 

0° 

1 

28" 

1'  23" 

1' 

9" 

O'  50" 

O' 32" 

O' 

18" 

30° 

1 

1 

28 

1 

23 

1 

8 

0 

49 

0 

31 

0 

18 

29 

2 

1 

28 

1 

22 

1 

8 

0 

49 

0  30 

0 

18 

28 

3 

1 

28 

1 

22 

1 

7 

0 

48 

0 

30 

0 

17 

27 

4. 

1 

28 

1 

22 

1 

7 

0 

47 

0 

29 

0 

17 

26 

5 

1 

28 

1 

21 

1 

6 

0 

47 

0 

29 

0 

17 

25 

6 

1 

28 

1 

21 

1 

5 

0 

46 

0 

28 

0 

17 

24 

7 

1 

28 

1 

20 

1 

5 

0 

46 

0 

28 

0 

16 

23 

8 

1 

28 

1 

20 

1 

4 

0 

45 

0 

27 

0 

16 

22 

9 

1 

28 

1 

20 

1 

4 

0 

44 

0 

27 

0 

16 

21 

10 

1 

28 

1 

19 

1 

3 

0 

44 

0 

26 

0 

16 

20 

11 

1 

28 

1 

19 

1 

2 

0 

43 

0 

26 

0 

15 

19 

12 

1 

27 

1 

18 

1 

2 

0 

42 

0 

25 

0 

15 

18 

13 

1 

27 

1 

18 

1 

1 

0 

42 

0 

25 

0 

15 

17 

14 

1 

27 

1 

17 

1 

0 

0 

41 

0 

24 

0 

15 

16 

15 

.1 

27 

1 

17 

1 

0 

0 

40 

0 

24 

0 

15 

15 

16 

1 

27 

1 

16 

0 

59 

0 

40 

0 

24 

0 

15 

14 

17 

1 

27 

1 

16 

0 

59 

0 

39 

0 

23 

0 

14 

13 

18 

1 

26 

1 

15 

0 

58 

0 

39 

0 

23 

0 

14 

12 

19 

1 

26 

1 

15 

0 

57 

0 

38 

0 

22 

0 

14 

11 

20 

1 

26 

1 

14 

0 

57 

0 

37 

0 

22 

0 

14 

10 

21 

1 

26 

1 

14 

0 

56 

0 

37 

0 

21 

0 

14 

9 

22 

1 

25 

1 

13 

0 

55 

0 

36 

0 

21 

0 

14 

8 

23 

1 

25 

1 

13 

0 

55 

0 

36 

0 

21 

0 

14 

7 

24 

1 

25 

1 

12 

0 

54 

0 

35 

0 

20 

0 

14 

6 

25 

1 

25 

1 

12 

0 

53 

0 

34 

0 

20 

0 

14 

5 

26 

1 

24 

1 

11 

0 

53 

0 

34 

0 

20 

0 

14 

4 

27 

1 

24 

1 

11 

0 

52 

0 

on 

JJ 

0 

19 

0 

14 

3 

28 

1 

24 

1 

10 

0 

51 

0 

33 

0 

19 

0 

13 

2 

29 

1 

23 

1 

10 

0 

51 

0 

32 

0 

19 

0 

13 

1 

30 

1 

23 

1 

9 

0 

50 

0 

32 

0 

18 

0 

13 

0 

XIs 

X* 

IX* 

VIII* 

VII* 

VI* 

TABLE  XL! 


53 


Moon's  Equatorial  Parallax. 
Argument*  Anomaly. 


0* 

Is 

il* 

III* 

1\ 

/« 

V 

i 

0° 

58' 

58" 

58' 

27" 

57' 

8" 

55' 

30" 

54' 

2" 

53' 

3" 

30° 

1 

58 

58 

58 

25 

57 

5 

55 

27 

53 

59 

53 

2 

29 

2 

58 

58 

58 

23 

57 

2 

55 

23 

53 

57 

53 

0 

28 

3 

58 

57 

58 

21 

56 

58 

55 

20 

53 

54 

52 

59 

27 

4 

58 

57 

58 

19 

56 

55 

55 

17 

53 

52 

52 

58 

26 

5 

58 

57 

58 

16 

56 

52 

55 

14 

53 

50 

52 

57 

25 

6 

58 

56 

58 

14 

56 

49 

55 

11 

53 

47 

52 

56 

24 

7 

58 

56 

58 

12 

56 

45 

55 

7 

53 

45 

52 

55 

23 

8 

58 

55 

58 

10 

56 

42 

55 

4 

53 

43 

52 

54 

22 

9 

58 

55 

58 

7 

56 

39 

55 

1 

53 

41 

52 

53 

21 

10 

58 

54 

58 

5 

56 

36 

54 

58 

53 

38 

52 

52 

20 

11 

58 

53 

58 

2 

56 

32 

54 

55 

53 

36 

52 

51 

19 

12 

58 

53 

58 

0 

56 

29 

54 

52 

53 

34 

52 

50 

18 

13 

58 

52 

57 

57 

56 

26 

54 

49 

53 

32 

52 

49 

17 

14 

58 

51 

57 

55 

56 

22 

54 

46 

53 

30 

52 

49 

16 

15 

58 

50 

57 

52 

56 

19 

54 

43 

53 

28 

52 

48 

15 

16 

58 

49 

57 

49 

56 

16 

54 

40 

53 

26 

52 

47 

14 

17 

58 

48 

57 

46 

56 

13 

54 

37 

53 

24 

52 

47 

13 

18 

58 

46 

57 

44 

56 

9 

54 

34 

53 

22 

52 

46 

12 

19 

58 

45 

57 

41 

56 

6 

54 

31 

53 

21 

52 

45 

11 

20 

58 

44 

57 

38 

56 

3 

54 

29 

53 

19 

52 

45 

10 

21 

58 

42 

57 

35 

55 

59 

54 

26 

53 

17 

52 

45 

9 

22 

58 

41 

57 

32 

55 

56 

54 

23 

53 

15 

52 

44 

8 

23 

58 

39 

57 

29 

55 

53 

54 

20 

53 

14 

52 

44 

7 

24 

58 

38 

57 

26 

55 

49 

54 

18 

53 

12 

52 

43 

6 

25 

58 

36 

57 

23 

55 

46 

54 

15 

53 

10 

52 

43 

5 

26 

58 

34 

57 

20 

55 

43 

54 

12 

53 

9 

52 

43 

4 

27 

58 

33 

57 

17 

55 

40 

54 

10 

53 

7 

52 

43 

3 

28 

58 

31 

57 

14 

55 

36 

54 

7 

53 

6 

52 

43 

2 

29 

58 

29 

57 

11 

55 

33 

54 

4 

53 

4 

52 

43 

1 

30 

58 

27 

57 

8 

55 

30 

54 

2 

53 

3 

52 

43 

0 

XI* 

X* 

IX* 

vm« 

VXI* 

VI* 

TABLE  XLII 


TABLE  XLIII 


45 


Moon's  Equatorial  Parallax . 
Argument.  Argument  of  the  Va¬ 
riation. 


Reduction  of  the  Parallax , 
and  also  of  the  Latitude . 
Argument.  Latitude. 


0* 

1* 

li* 

III* 

IV* 

V* 

0° 

56" 

42" 

16" 

4" 

18" 

44" 

30° 

1 

56 

41 

15 

4 

18 

45 

29 

2 

55 

41 

14 

4 

19 

46 

28 

3 

55 

40 

14 

4 

20 

46 

27 

4 

55 

39 

13 

4 

21 

47 

26 

5 

55 

38 

12 

4 

22 

48 

25 

6 

55 

37 

12 

4 

23 

48 

24 

r 

55 

36 

11 

5 

24 

49 

23 

8 

55 

35 

10 

5 

24 

50 

22 

9 

54 

35 

10 

5 

25 

50 

21 

10 

54 

34 

9 

6 

26 

51 

20 

11 

54 

33 

9 

6 

27 

51 

19 

12 

53 

32 

8 

6 

28 

52 

18 

13 

53 

31 

8 

7 

29 

53 

17 

14 

52 

30 

7 

7 

30 

53 

16 

15 

52 

29 

7 

8 

31 

53 

15 

16 

51 

28 

6 

8 

32 

54 

14 

17 

51 

27 

6 

9 

33 

54 

13 

18 

50 

26 

6 

9 

34 

55 

12 

19 

50 

25 

5 

10 

35 

55 

11 

20 

49 

24 

5 

10 

35 

55 

10 

21 

49 

24 

5 

11 

36 

56 

9 

22 

48 

23 

4 

12 

37 

56 

8 

23 

47 

22 

4 

12 

38 

56 

7 

24 

47 

21 

4 

13 

39 

56 

6 

25 

46 

20 

4 

14 

40 

57 

5 

26 

45 

19 

4 

14 

41 

57 

4 

27 

45 

18 

4 

15 

42 

57 

3 

28 

44 

18 

4 

16 

42 

57 

2 

29 

43 

17 

4 

17 

43 

57 

1 

30 

42 

16 

4 

18 

44 

57 

0 

XI* 

X* 

IXs 

VIII* 

VII* 

i  vp 

Lat. 

Red.  of 
Paral. 

Red.  of 
Lat. 

0° 

0' 

O' 

0" 

3 

0 

1 

12 

6 

0 

2 

23 

9 

0 

3 

32 

12 

0 

4 

39 

15 

1 

5 

43 

18 

1 

6 

44 

21 

1 

7 

40 

24 

2 

8 

31 

27 

'  2 

9 

16 

30 

3 

9 

55 

33 

3 

10 

28 

36 

4 

10 

54 

39 

5 

11 

13 

42 

5 

11 

25 

45 

6 

11 

29 

48 

6 

11 

25 

51 

7 

11 

14 

54 

8 

10 

56 

57 

8 

10 

30 

60 

9 

9 

57 

63 

9 

9 

18 

66 

10 

8 

33 

69 

10 

7 

42 

72 

10 

6 

46 

75 

11 

5 

45 

78 

11 

4 

41 

81 

11 

3 

33 

84 

11 

2 

24 

87 

11 

1 

12 

90 

11 

,  0 

0 

TABLE  XLI V. 


Moon's  Semidiameter. 
Argument.  Equatorial  Parallax. 


Eq. 

Par. 

Semidi.tEq. 

Par. 

Semidi. 

Eq. 

Par. 

Semidi. 

53 

0 " 

14' 

27" 

56' 

0" 

15 

16" 

59' 

'  0" 

16' 

5" 

53 

10 

14 

29 

56 

10 

15 

18 

59 

10 

16 

7 

53 

20 

14 

32 

56 

20 

15 

21 

59 

20 

16 

10 

53 

30 

14 

35 

56 

30 

15 

24 

59 

30 

16 

13 

53 

40 

14 

37 

56 

40 

15 

26 

59 

40 

16 

16 

53 

50 

14 

40 

56 

50 

15 

29 

59 

50 

16 

18 

54 

0 

14 

43 

57 

0 

15 

32 

60 

0 

16 

21 

54 

10 

14 

46 

57 

10 

15 

35 

60 

10 

16 

24 

54 

20 

14 

48 

57 

20 

15 

37 

60 

20 

16 

26 

54 

30 

14 

51 

57 

30 

15 

40 

60 

30 

16 

29 

54 

40 

14 

54 

57 

40 

15 

43 

60 

40 

16 

32 

54 

50 

14 

57 

57 

50 

15 

46 

60 

50 

16 

35 

55 

0 

14 

59 

58 

0 

15 

48 

61 

0 

16 

37 

55 

10 

15 

2 

58 

10 

15 

51 

61 

10 

16 

40 

55 

20 

15 

5 

58 

20 

15 

54 

61 

20 

16 

43 

55 

30 

15 

7 

58 

30 

15 

56 

61 

30 

16 

46 

55 

40 

15 

10 

58 

40 

15 

59 

61 

40 

16 

48 

55 

50 

15 

13 

58 

50 

16 

2 

61 

50 

16 

51 

56 

0 

15 

16 

59 

0 

16 

5 

62 

0 

16 

54 

TABLE  XLV. 

Augmentation  of  Moon's 
Semidiameter. 

Argum.  Appar.  Alt. 


TABLE  XLVI. 

Moon's  Horary  Motion  in 
Longitude . 

Arguments.  2,  3,  4  and 
5  of  Long. 


Arg. 

2 

3 

4 

5 

Arg. 

0 

6" 

1" 

3" 

3" 

100 

5 

5 

2 

3 

3 

95 

10 

5 

2 

3 

3 

90 

15 

4 

2 

3 

3 

85 

20 

4 

3 

2 

2 

80 

25 

3 

3 

2 

2 

75 

30 

2 

3 

2 

2 

70 

35 

2 

4 

1 

1 

65 

40 

1 

4 

1 

1 

60 

45 

1 

4 

1 

1 

55 

50 

0 

5 

1 

1 

50 

Ap.  Alt 

Augm. 

6° 

2" 

12 

3 

18 

5 

24 

6 

30 

8 

36 

9 

42 

11 

48 

12 

54 

13 

60 

14 

66 

15 

72 

15 

78 

16 

84 

16 

90 

16 

56 


TABLE  XLVII, 


Moon's  Horary  Motion  in  Longitude . 
Argument.  Argument  of  the  Evection. 


0* 

I* 

II* 

III* 

IV* 

1 

0° 

V  20" 

V 

15" 

V 

0" 

O' 39" 

O' 20" 

O' 6" 

30° 

1 

1 

20 

1 

14 

0 

59 

0  39 

0 

19 

0 

6 

29 

2 

1 

20 

1 

14 

0 

58 

0  38 

0 

19 

0 

5 

28 

3 

1 

20 

1 

14 

0 

58 

0 

37 

0 

18 

0 

5 

27 

4 

1 

20 

1 

13 

0 

57 

0 

37 

0 

18 

0 

5 

26 

5 

1 

20 

1 

13 

0 

56 

0 

36 

0 

17 

0 

4 

25 

6 

1 

20 

1 

12 

0 

56 

0 

35 

0 

16 

0 

4 

24 

7 

1 

20 

1 

12 

0 

55 

0 

35 

0 

16 

0 

4 

23 

8 

1 

20 

1 

11 

0 

54 

0 

34 

0 

15 

0 

4 

22 

9 

1 

20 

1 

11 

0 

54 

0 

33 

0 

15 

0  3 

21 

10 

1 

20 

1 

11 

0 

53 

0 

33 

0 

14 

0  3 

20 

11 

1 

20 

1 

10 

0 

52 

0 

32 

0 

14 

0 

3 

19 

12 

1 

19 

1 

10 

0 

52 

0 

31 

0 

13 

0 

3 

18 

13 

1 

19 

1 

9 

0 

51 

0 

31 

0 

13 

0 

3 

17 

14 

1 

19 

1 

9 

0 

50 

0 

30 

0 

12 

0 

2 

16 

15 

1 

19 

1 

8 

0 

50 

0 

29 

0 

12 

0 

2 

15 

16 

1 

19 

1 

8 

0 

49 

0 

29 

0 

11 

0 

2 

14 

17 

1 

18 

1 

7 

0 

48 

0 

28 

0 

11 

0 

2 

13 

18 

1 

18 

1 

7 

0 

48 

0 

27 

0 

11 

0 

2 

12 

19 

1 

18 

1 

6 

0 

47 

0 

27 

0 

10 

0 

2 

11 

20 

1 

18 

1 

5 

0 

46 

0 

26 

0 

10 

0 

1 

10 

21 

1 

18 

1 

5 

0 

46 

0 

25 

0 

9 

0 

1 

9 

22 

1 

17 

1 

4 

0 

45 

0 

25 

0 

9 

0 

1 

8 

23 

1 

17 

1 

4 

0 

44 

0 

24 

0 

8 

0 

1 

7 

24 

1 

17 

1 

3 

0 

44 

0 

23 

0 

8 

0 

1 

6 

25 

1 

16 

1 

3 

0 

43 

0  23 

0 

8 

0 

1 

•  5 

26 

1 

16 

1 

2 

0  42 

0 

22 

0 

7 

0 

1 

4 

27 

1 

16 

1 

1 

0  41 

0 

22 

0 

7 

0 

1 

3 

28 

1 

15 

1 

1 

0  41 

0 

21 

0 

7 

0 

1 

2 

29 

1 

15 

1 

0 

0 

40 

0 

20 

0 

6 

0 

1 

1 

30 

1 

15 

1 

0 

0 

39 

0 

20 

0 

6 

0 

1 

0 

XL* 

1  X* 

IX* 

Mil* 

VII* 

VI* 

TABLE  XLVIII, 


57 


Moon's  Horary  Motion  in  Longitude. 

Arguments.  Sum  of  preceding  equations,  and  Anomaly,  cor¬ 
rected. 


0" 

10'' 

20" 

30" 

40" 

50" 

60" 

70" 

80" 

90" 

100" 

0*  0° 

4" 

5" 

6" 

8" 

9" 

10" 

11" 

12" 

14" 

15" 

16" 

XIIs  0° 

5 

4 

5 

6 

8 

9 

10 

il 

12 

14 

15 

16 

25 

10 

4 

5 

7 

8 

9 

10 

11 

12 

13 

15 

16 

20 

15 

4 

5 

7 

8 

9 

10 

11 

12 

13 

15 

16 

15 

20 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

10 

25 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

5 

I  0 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

XI  0 

5 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

25 

10 

6 

7 

7 

8 

9 

10 

11 

12 

13 

13 

14 

20 

15 

6 

7 

8 

8 

9 

10 

11 

12 

12 

13 

14 

15 

20 

7 

7 

8 

9 

9 

10 

11 

11 

12 

13 

13 

10 

25 

7 

8 

8 

9 

9 

10 

11 

11 

12 

12 

13 

5 

II  0 

7 

8 

8 

9 

9 

10 

11 

11 

12 

12 

13 

X  0 

5 

8 

8 

9 

9 

lo 

10 

10 

11 

11 

12 

12 

25 

10 

8 

9 

9 

9 

10 

10 

10 

11 

11 

11 

12 

20 

15 

9 

9 

9 

1G 

10 

10 

10 

10 

11 

11 

11 

15 

20 

9 

10 

10 

10 

10 

10 

10 

10 

10 

10 

11 

10 

25 

10 

10 

10 

10 

10 

10 

10 

10 

10 

10 

10 

5 

III  0 

10 

lu 

10 

lu 

10  ; 

10 

10 

10  i 

10 

10 

10 

IX  0 

5 

11 

11 

11 

10 

10 

10 

10 

10 

9 

9 

9 

25 

10 

11 

11 

11 

11 

10 

10 

10 

9 

9 

9 

9 

20 

15 

12 

11 

11 

11 

10 

10 

10 

9 

9 

9 

8 

15 

20 

12 

12 

11 

11 

10 

10 

10 

9 

9 

8 

8 

10 

25 

13 

12 

12 

11 

11 

10 

9 

9 

8 

8 

7 

5 

IV  0 

13 

12 

12 

11 

11 

10 

9 

9 

8 

8 

7 

VIM  0 

5 

13 

13 

12 

11 

11 

10 

9 

9 

8 

7 

7 

25 

10 

14 

13 

12 

11 

11 

10 

9 

9 

8 

7 

6 

20 

15 

14 

13 

12 

12 

11 

10 

9 

8 

8 

7 

6 

15 

20 

14 

13 

12 

12 

11 

10 

9 

8 

8 

7 

6 

10 

25 

14 

13 

13 

12 

11 

10 

9 

8 

7 

7 

6 

5 

V  0 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

VII  0 

5 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

25 

10 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

20 

15 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

15 

20 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

10 

25 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

5 

vr  o 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

VI  0 

1 

0" 

10" 

20" 

30" 

40" 

50" 

60" 

70" 

80' 

90" 

10  0" 

8* 


58 


TABLE  XL1X 


Moon's  Horary  Motion  in  Longitude. 
Argument.  Anomaly,  corrected. 


0’ 

1* 

IIs 

m. 

IVs 

V» 

0° 

34' 

51" 

34/ 

14" 

32' 39" 

30' 

45" 

29' 

6" 

28' 

1" 

30° 

1 

34 

51 

34 

12 

32 

36 

30 

42 

29 

3 

27 

59 

29 

2 

34 

51 

34 

9 

32 

32 

30 

38 

29 

0 

27 

58 

28 

3 

34 

51 

34 

7 

32 

28 

30 

34 

28 

58 

27 

56 

27 

4 

34 

51 

34 

4 

32 

24 

30 

31 

28 

55 

27 

55 

26 

5 

34 

50 

34 

1 

32 

21 

30 

27 

28 

52 

27 

54 

25 

6 

34 

50 

33 

59 

32 

17 

30 

23 

28 

50 

27 

53 

24 

7 

34 

49 

33 

56 

32 

13 

30 

20 

28 

47 

27 

51 

23 

8 

34 

49 

33 

53 

32 

9 

30 

16 

28 

45 

27 

50 

22 

9 

34 

48 

33 

50 

32 

5 

30 

13 

28 

42 

27 

49 

21 

10 

34 

47 

33 

47 

32 

2 

30 

9 

28 

40 

27 

48 

20 

11 

34 

46 

33 

44 

31 

58 

30 

6 

28 

37 

27 

47 

19 

12 

34 

45 

33 

41 

31 

54 

30 

2 

28 

35, 

27 

46 

18 

13 

34 

44 

33 

38 

31 

50 

29 

59 

28 

33 

27 

45 

17 

14 

34 

43 

33 

35 

31 

46 

29 

56 

28 

30 

27 

45 

16 

15 

34 

42 

33 

32 

31 

42 

29 

52 

28 

28 

27 

44 

15 

16 

34 

41 

33 

28 

31 

38 

29 

49 

28 

26 

27 

43 

14 

17 

34 

39 

33 

25 

31 

35 

29 

46 

28 

24 

27 

42 

13 

18 

34 

38 

33 

22 

31 

31 

29 

42 

28 

22 

27 

42 

12 

19 

34 

36 

33 

18 

31 

27 

29 

39 

28 

20 

27 

41 

11 

20 

34 

34 

33 

15 

31 

23 

29 

36 

28 

18 

27 

41 

10 

21 

34 

33 

33 

12 

31 

19 

29 

33 

28 

16 

27 

40 

9 

22 

34 

31 

33 

8 

31 

15 

29 

30 

28 

14 

27 

40 

8 

23 

34 

29 

33 

5 

31 

12 

29 

26 

28 

12 

27 

39 

7 

24 

34 

27 

33 

1 

31 

8 

29 

23 

28 

10 

27 

39 

6 

25 

34 

25 

32 

58 

31 

4 

29 

20 

28 

9 

27 

39 

5 

26 

34 

23 

32 

54 

31 

0 

29 

17 

28 

7 

27 

39 

4 

27 

34 

21 

32 

50 

30 

57 

29 

14 

28 

5 

27 

38 

3 

28 

34 

19 

32 

47 

30 

53 

29 

12 

28 

4 

27 

38 

2 

29 

34 

16 

32 

43 

30 

49 

29 

9 

28 

2 

27 

38 

1 

30 

34 

14 

32 

39 

30 

45 

29 

6 

28 

1 

27 

38 

0 

XIs 

Xs 

IXs 

VIIIs 

VIIs 

VIs 

TABLE  l 


59 


Moon’s  Horary  Motion  in  Longitude. 

Arguments.  Sum  of  preceding  equations,  and  Arg.  of  Variation. 


to 

28' 

29' 

30' 

31' 

32' 

33' 

34' 

35' 

36' 

37' 

0*  0° 

0" 

1" 

2" 

4" 

5" 

6" 

7" 

8" 

10" 

11" 

12" 

XII*  0° 

5 

0 

1 

2 

4 

5 

6 

7 

8 

10 

11 

12 

25 

10 

0 

1 

3 

4 

5 

6 

7 

8 

9 

11 

12 

20 

15 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

15 

20 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

10 

25 

2 

3 

4 

4 

5 

6 

7 

8 

8 

9 

10 

5 

I  0 

3 

4 

4 

5 

5 

6 

7 

7 

8 

8 

9 

XI  0 

5 

4 

4 

5 

5 

6 

6 

6 

7 

7 

8 

8 

25 

10 

5 

5 

5 

6 

6 

6 

6 

6 

7 

7 

7 

20 

15 

6 

6 

6 

6 

6 

6 

6 

6 

6 

6 

6 

15 

20 

7 

7 

7 

7 

6 

6 

6 

5 

5 

5 

5 

10 

25 

8 

8 

7 

7 

6 

6 

6 

5 

5 

4 

4 

5 

II  0 

9 

9 

8 

7 

7 

6 

5 

5 

4 

3 

3 

X  0 

5 

10 

9 

8 

8 

7 

6 

5 

4 

4 

3 

2 

25 

10 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

20 

15 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

15 

20 

12 

11 

10 

8 

7 

6 

5 

4 

2 

1 

0 

10 

25 

12 

11 

10 

8 

7 

6 

5 

4 

2 

1 

0 

5 

III  0 

12 

11 

10 

8 

7 

6 

5 

4 

2 

1 

0 

IX  0 

5 

12 

11 

10 

8 

7 

6 

5 

4 

2 

1 

0 

25 

10 

12 

11 

10 

8 

7 

6 

5 

4 

2 

1 

0 

20 

15 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

15 

20 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

10 

25 

10 

9 

8 

8 

7 

€ 

5 

4 

4 

3 

2 

5 

IV  0 

9 

8 

8 

7 

7 

6 

5 

5 

4 

4 

3 

VIII  0 

5 

8 

8 

7 

7 

6 

6 

6 

5 

5 

4 

4 

25 

10 

7 

7 

7 : 

6 

6 

6 

6 

6 

5 

5 

5 

20 

15 

6 

6 

6 

6 

6 

6 

6 

6 

6 

6 

6 

15 

20 

5 

5 

5 

6 

6 

6 

6 

6 

7 

7 

7 

10 

25 

4 

4 

5 

5 

6 

6 

6 

7 

7 

8 

8 

5 

V  0 

3 

3 

4 

5 

5 

6 

7 

7 

8 

9 

9 

VII  0 

5 

2 

3 

3 

4 

5. 

6 

7 

8 

9 

9 

10 

25 

10 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

20 

15 

0 

2 

3 

4 

5 

6 

7 

8 

9 

10 

12 

15 

20 

0 

1 

2 

4 

5 

6 

7 

8 

10 

11 

12 

10 

25 

0 

1 

2 

3 

5 

6 

7 

9 

10 

11 

12 

5 

VI  0 

0 

1 

2 

3 

5 

6 

7 

9 

10 

11 

12 

VI  0 

. 

27' 

28' 

29' 

30' 

31' 

32' 

33' 

34' 

35' 

36' 

37' 

TABLE  LI 


Moon's  Horary  Motion  in  Longitude . 
Argument.  Argument  of  the  Variation. 


0s 

I* 

IIs 

m* 

IV*  1 

\ 

ft 

0° 

1' 

17" 

0' 

58": 

O'  20" 

O' 

2" 

O'  22" 

1' 

0" 

30° 

1 

1 

17 

0 

57 

0 

19 

0 

2 

0 

23 

1 

1 

29 

2 

1 

17 

0 

55 

0 

18 

0 

3 

0 

24 

1 

2 

28 

3 

1 

17 

0 

54 

0 

17 

0 

3 

0 

25 

1 

3 

27 

4 

1 

17 

0 

53 

0 

16 

0 

3 

0 

26 

1 

4 

26 

5 

1 

17 

0 

52 

0 

15 

0 

3 

0 

27 

1 

5 

25 

6 

1 

16 

0 

51 

0 

14 

0 

3 

0 

29 

1 

6 

24 

7 

1 

16 

0 

49 

0 

13 

0 

4 

i  0 

30 

1 

7 

23 

8 

1 

16 

0 

48 

0 

12 

0 

4 

:  0 

31 

1 

8 

22 

9 

1 

15 

0 

47 

0 

11 

0 

4 

0 

33 

1 

9 

21 

10 

1 

15 

0 

45 

0 

11 

0 

5 

0 

34 

1 

10 

20 

11 

1 

14 

0 

44 

0 

10 

0 

5 

0 

35 

1 

11 

19 

12 

1 

14 

0 

43 

0 

9 

0 

6 

0 

37 

1 

12 

18 

13 

1 

13 

0 

41 

0 

8 

0 

6 

0 

38 

1 

13 

17 

14 

1 

13 

0 

40 

0 

8 

0 

7 

0 

39 

1 

13 

16 

15 

1 

12 

0 

39 

0 

7 

0 

8 

0 

40 

1 

14 

15 

16 

1 

11 

0 

38 

0 

6 

0 

8 

0 

42 

1 

1j 

14 

17 

1 

11 

0 

36 

0 

6 

0 

9 

0 

43 

1 

15 

13 

18 

1 

10 

0 

35 

0 

5 

0 

10 

0 

44 

1 

16 

12 

19 

1 

9 

0  34 

0 

5 

0 

11 

0 

46 

1 

16 

11 

20 

1 

8 

0 

32 

0 

4 

0 

11 

0 

47 

1 

17 

10 

21 

1 

7 

0 

31 

0 

4 

0 

12 

0 

48 

1 

17 

9 

22 

1 

6 

0 

30 

0 

4 

0 

13 

0 

50 

1 

18 

8 

23 

1 

5 

0 

29 

0 

3 

0 

14 

0 

51 

1 

18 

7 

24 

1 

4 

0 

27 

0 

3 

0 

15 

0 

52 

1 

18 

6 

25 

1 

3 

0 

26 

0 

3 

0 

16 

0 

54 

1 

19 

5 

26 

1 

2 

0 

25 

0 

3 

0 

17 

0 

55 

1 

19 

4 

27 

1 

1 

0 

24 

0 

3 

0 

18 

0 

56 

1 

19 

3 

28 

1 

0 

0 

23 

0 

2 

0 

19 

0 

57 

1 

19 

2 

29 

0  39 

0 

21 

0 

2 

0  20 

0 

59 

1 

19 

1 

30 

0 

58 

0 

20 

0 

2 

0 

22 

1 

0 

1 

19 

0 

' 

XIs 

1  X* 

IX* 

VIII* 

VII* 

VI* 

TABLE  LH 


61 


Moon's  Horary  Motion  in  Longitude . 
Argument.  Argument  of  the  Reduction. 


0* 

1* 

11* 

III* 

IV* 

V* 

0° 

2" 

6" 

14" 

18" 

14" 

6" 

30° 

1 

2 

6 

14 

18 

14 

6 

29 

2 

2 

7 

14 

18 

13 

6 

28 

3 

2 

7 

15 

18 

13 

5 

27 

4 

2 

7 

15 

18 

13 

5 

26 

5 

2 

7 

15 

18 

13 

5 

25 

6 

2 

8 

15 

18 

12 

5 

24 

7 

2 

8 

16 

18 

12 

4 

23 

8 

2 

8 

16 

18 

12 

4 

22 

9 

2 

8 

16 

18 

12 

4 

21 

10 

3 

9 

16 

17 

11 

4 

20 

11 

3 

9 

16 

17 

11 

4 

19 

12 

3 

9 

16 

17 

11 

4' 

18 

13 

3 

9 

17 

17 

11 

3 

17 

14 

3 

10 

17 

17 

10 

3 

16 

15 

3 

10 

17 

17 

10 

3 

15 

16 

3 

10 

17 

17 

10 

3 

14 

17 

3 

11 

17 

17 

9 

3 

13 

18 

4 

11 

17 

16 

9 

3 

12 

19 

4 

11 

17 

16 

9 

3 

11 

20 

4 

11 

17 

16 

9 

3 

10 

21 

4 

12 

18 

16 

8 

2 

9 

22 

4 

12 

18 

16 

8 

2 

8 

23 

4 

12  i 

18 

16 

8 

2 

7 

24 

5 

12 

18 

15 

8 

2 

6 

25 

5 

13 

18 

15 

7 

2 

5 

26 

5 

13 

18 

15 

7 

2 

4 

27 

5 

13 

18 

15 

7 

2 

3 

28 

6 

13 

18 

14 

7 

2 

2 

29 

6 

14 

18 

14 

6 

2 

1 

30 

6 

14 

18 

14 

6 

2 

0 

XI* 

X* 

IX* 

Vlil* 

VII* 

VI* 

/ 


62 


TABLE  L11I, 


Moon's  Horary  Motion  in  Latitude . 
Argument.  Arg.  I,  of  Latitude. 


0*  + 

I'  + 

1U  + 

III*  — 

IV*— 

V*  — 

0° 

2'  58" 

2 

34" 

V  29" 

O' 

0" 

V 

29" 

2'  34" 

30° 

1 

2 

58 

2 

33 

1 

27 

0 

3 

1 

32 

2 

36 

29 

2 

2 

58 

2 

31 

1 

24 

0 

6 

1 

35 

2 

37 

28 

3 

2 

58 

2 

29 

1 

21 

0 

9 

1 

37 

2 

39 

27 

4 

2 

58 

2 

28 

1 

18 

0 

12 

1 

40 

2 

40 

26 

5 

2 

57 

2 

26 

1 

15 

0 

16 

1 

42 

2 

41 

25 

6 

2 

57 

2 

24 

1 

13 

0 

19 

1 

45 

2 

43 

24 

7 

2 

57 

2 

22 

1 

10 

0 

22 

1 

47 

2 

44 

23 

8 

2 

56 

2 

20 

1 

7 

0 

25 

1 

50 

2 

45 

22 

9 

2 

56 

2 

19 

1 

4 

0 

28 

1 

52 

2 

46 

21 

10 

2 

55 

2 

17 

1 

1 

0 

31 

1 

55 

2 

47 

20 

11 

2 

55 

2 

15 

0 

58 

0 

34 

1 

57 

2 

48 

19 

12 

2 

54 

2 

12 

0 

55 

0 

37 

1 

59 

2 

49 

18 

13 

2 

53 

2 

10 

0 

52 

0 

40 

2 

2 

2 

50 

17 

14 

2 

53 

2 

8 

0 

49 

0 

43 

2 

4 

2 

51 

16 

;  15 

2 

52 

2 

6 

0 

46 

0 

46 

2 

6 

2 

52 

15 

16 

2 

51 

2 

4 

0 

43 

0 

49 

2 

8 

2 

53 

14 

17 

2 

50 

2 

2 

0 

40 

0 

52 

2 

10 

2 

53 

13 

18 

2 

49 

1 

59 

0 

37 

0 

55 

2 

12 

2 

54 

12 

19 

2 

48 

1 

57 

0 

34 

0 

58 

2 

15 

2 

55 

11 

20 

2  47 

1 

55 

0 

31 

1 

1 

2 

17 

2 

55 

10 

21 

2 

46 

1 

52 

0 

28 

1 

4 

2 

19 

2 

56 

9 

22 

2 

45 

1 

50 

0 

25 

1 

7 

2 

20 

2 

56 

8 

23 

2 

44 

1 

47 

0 

22- 

1 

10 

2 

22 

2 

57 

7 

24 

2 

43 

1 

45 

0 

19 

1 

13 

2 

24 

2 

57 

6 

25 

2 

41 

1 

42 

0 

16 

1 

15 

2 

26 

2 

57 

5 

26 

2 

40 

1 

40 

0 

12 

1 

18 

2 

28 

2 

58 

4 

27 

2 

39 

1 

37 

0 

9 

1 

21 

2 

29 

2 

58 

3 

28 

2 

37 

1 

35 

0 

6 

1 

24 

2 

31 

2 

58 

2 

29 

2 

36 

1 

32 

0 

3 

1 

27 

2 

33 

2 

58 

1 

30 

2 

34 

1 

29 

0 

0 

1 

29 

2 

34 

2 

58 

0 

XI»+ 

x*  + 

IX*+ 

VIII*— 

VII*— 

VI*- 

TABLE  LIV. 


Moon's  Horary  Motion  in  Latitude . 
Argument.  Arg.  II,  of  Latitude. 


o  + 

1*  + 

1I*+ 

111*  — 

IV*— 

V*  — 

0° 

4" 

4" 

2" 

0" 

2" 

4" 

30° 

6 

4 

3 

2 

0 

3 

4 

24 

12 

4 

3 

1 

1 

3 

4 

18 

18 

4 

3 

1 

1 

3 

4 

12 

24 

4 

3 

0 

2 

3 

4 

6 

30 

4 

2 

0 

2 

4 

4 

0 

XI* + 

x*+ 

IX*-f- 

vrn*  — 

VII* 

VI*— 

TABLE  LV, 


63 


Nautical  Almanac. 


I.  AUGUST  1821. 


r^ 

Phases  of  the  Moon. 

0) 

c 

£ 

_ 

<y 

CD 

Sundays,  Holidays , 

Terms ,  &c. 

D  H.  M. 

o 

O 

*3)  First  Quarter ,  5.14.10 

CO 

>> 

O  Full  Moon,  13.  2.  8 
(£  Last  Quarter,  19.18.49 

Q 

Q 

•  New  Moon,  27.  3.17 

w. 

Th. 

1 

2 

Lammas-Day. 

Other  Phenomena. 

F. 

3 

D.  H.  M. 

Sa. 

4 

3.17.47  J)  a  r% 

10  -  -  lj  Stationary. 

Sun. 

5 

7/A  Sunday  after  Trinity. 

11  -  -  $  Stationary. 

M.  . 

6 

Transfig.  of  our  Lord. 

19  -  -  2/  Stationary. 

Tu. 

7 

Name  of  Jesus.  Princess 

23.  1.  1  0  enters 

W. 

8 

[ Amelia  born. 

25.17.  0  D  $ 

Th. 

9 

26.14.30  D  a  SI 

F. 

10 

St.  Lawrence. 

27  -  -  O  eclipsed  in  vis. 

Sa. 

11 

Prs.  of  JSninsTvick  bom. 

21.11.58  D  /3  8 

31.  1.32  D  a  11% 

Sun. 

12 

8th  Sunday  aft.  Trinity. 

M. 

13 

Prince  of  IV ales  b. 

Tu, 

14 

W. 

15 

Assumption. 

Th. 

16 

Duke  of  York  bom. 

F. 

17 

Sa. 

18 

Sun. 

19 

9th  Sunday  aft.  Tnnity. 

M. 

20 

Tu. 

21- 

Duke  of  Clarence  bom. 

W. 

22 

Th. 

23 

F. 

24 

St.  Bartholomew. 

Sa. 

25 

Sun. 

26 

10/A  Sunday  aft.  Trinity. 

M. 

27 

Tu. 

28 

St.  Augustine. 

* 

W. 

29 

St.  John  Bapt. beheaded. 

Th. 

30 

F. 

31 

64 


TABLE  LV. 


Nautical  Almanac. 


AUGUST  1821.  II. 


Days  of  the  Week. 

Days  of  the  Month. 

THE  SUN’S 

Equation 

ofTime, 

Add. 

I 

DiflT. 

Longitude. 

Right  Ascen. 
in  Time. 

jDeclination 

North. 

S.  D.  M.  S. 

H.  M  S. 

D.  M.S. 

M.  S. 

S. 

W. 

Th. 

F. 

Sa. 

Sun. 

1 

2 

3 

4 

5 

4.  8.49.37 
4.  9.47.  5 
4.10.44.33 
4.11.42-  2 
4.12.39.32 

8.45.  2,9 

8.48.55.9 
8.52.48  2 

8.56.39.9 
9.  0.31,0 

18.  4.20 
17.49.  6 
17.33.35 
17.17.46 
17.  1.41 

5.58.4 
5.548 

5.50.5 
5.45  7 
5.40,3 

3.6 
43 

4.8 
5,4 

6,1 

6.7 

7.3 

7. 9 

8.4 

9,1 

9.7 
10,2 

10.7 

11.3 

11.8 

12.3 
12,8 

13.3 

13.7 

14.2 

14.6 

15.1 
15,5 

15.8 

16.3 

16.7 
17,0 

17.4 

17.8 

18.2 

M. 

Tu. 

W. 

Th. 

F. 

6 

7 

8 

9 

10 

4.13.37.  2 
4.14.34.34 
4.15.32.  6 
4.16.29.39 
4.17.27.12 

9.  4.21,5 
9.  8.11,3 
9.12.  0,6 

9.15.49.2 

9.19.37.3 

16.45.19 

16.28.41 

16.11.47 

15.54.37 

15.37.13 

5.34.2 
5.27,5 

5.20.2 

5.12.3 
5.  3,9 

Sa. 

Sun. 

M. 

Tu. 

W. 

11 

12 

13 

14 

15 

4.18.24.47 
4.19.22.23 
4.20.20.  0 
4.21.17.39 
4.22.15.18 

9.23.24,7 

9.27.11.6 
9.30.58,0 

9.34.43.7 
9.3 8.29,0 

15.19.33 
15.  1.39 
14.43.30 
14.25.  7 
14.  6.30 

4.54.8 

4.45.1 

4.34.9 

4.24.2 

4.12.9 

Th. 

F. 

Sa. 

Sun. 

M. 

16 

17 

18 

19 

20 

4.23.12.59 

4.24.10.42 

4.25.  8.26 

4.26.  6.12 

4.27.  4.  0 

9.42.13,7 

9.45.57.9 
9.49.41,6 

9.53.24.9 
9.57.  7,6 

13.47.40 
13.28.37 
13.  9.21 
12.49.51 
12.30.10 

4.  1,1 
3.48,8 
3.36,0 
3.22,7 

3.  9,0 

Tu. 

W. 

Th. 

F. 

Sa. 

21 

22 

23 

24 

25 

4.28.  1.49 
4.28.59.41 
4.29.57.34 
5.  0.55.29 
5.  1.53.26 

10.  0.50,0 
10.  4.31,9 
10.  8.13,3 
10.11.54,4 
10.15.35,0 

12.10.17 
11.50.11 
11.29.55 
11.  9.27 
10.48.49 

2.54,8 

2.40,2 

2.25,1 

2.  9,6 
1.53  8 

Sun. 

M. 

Tu. 

W. 

Th. 

26 

27 

28 

29 

30 

5.  2.51.25 
5.  3.49.25 
5.  4.47.26 
5.  5.45.30 
5.  6.43.34 

10.19.15.2 
10.22.55,1 

10.26.34.5 

10.30. 13.6 

10.33.52.3 

10.28.  0 
10.  7.  2 
9.45.53 
9.24.36 
9.  3.  9 

1.37,5 

1.20,8 

1.  3,8 
0.46  4 
0.28,6 

1  F. 

31 

5.  7.41.41 

10.37.30,7 

8.41.34 

0.10,4 

TABLE  LV, 


65 


Nautical  Almanac. 
III.  AUGUST  1821. 


Days. 

Time  of  Sun’s 
Semidiameter 
passingMerid. 

THE  SUN’S 

Place  of  the 
Moon’s 
Node . 

Semidi¬ 

ameter. 

H  ourly 
Motion. 

Logar. 

Distance. 

M.  S. 

M.  S. 

M.  S. 

S.  D.  M 

1 

7 

13 

19 

25 

1.  6,4 

1.  5,9 

1,  5,5 

1.  5,0 

1.  4,6 

15.47,5 

15.48.4 

15.49.4 

15.50.5 
15.51,7 

2.23,6 

2.23.9 
2.24,2 
2.24,5 

2.24.9 

0.00620 

0,00581 

0 .00536 
0.00485 
0.00429 

11.  5.43 
11.  5.24 
11.  5.  5 
11.  4.46 
11.  4.27 

ECLIPSES  OF  THE  SATELLITES  OF  JUPITER. 
MEAN  TIME. 


I. 

Satellite. 

II. 

Satellite. 

III.  Satellite. 

Irmnei'sions. 

Immersions. 

Days. 

H.  M.  S. 

Days. 

H.  M.  S. 

Days. 

H  M-  S. 

*2 

13.43.56 

*4 

11.  3.24 

*6 

12.  7.12  Im. 

4 

8.12.22 

8 

.0.22.37 

*6 

14.31.28  E. 

6 

2.40.46 

*11 

13.40.50 

13 

16.  8.10  Im. 

7 

21.  9.13 

15 

3.  0.  1 

13 

18.31.27  E. 

9 

15.37.36 

18 

16.18.11 

20 

20.  9.30  Im. 

11 

10.  6.  4 

22 

5.37.21 

20 

22.31.51  E. 

13 

4.34.28 

25 

18.55.31 

28 

0.10.  7  Im. 

14 

23.  2.55 

29 

8.14.40 

28 

2.31.36  E. 

16 

17.31.19 

*18 

11.59.48 

20 

6.28.13 

22 

0.56.41 

23 

19.25.  7 

*25 

13.53.36 

27 

8.22.  1 

29 

2.50.32 

30 

21.18.58 

1 

1 

IV.  Satellite. 

*9 


66 


TABLE  LV 


•Nautical  Almanac . 


AUGUST  1821.  IV. 


THE  PLANETS. 

Heliocentric. 

Geocentric. 

Declin. 

Rt.  asc. 

Passage 

t/5 

a 

P 

Long. 

Lat. 

Long. 

Lat. 

in  time. 

Merid. 

S.  D..M. 

D.  M. 

S.  D.  M. 

D.  M. 

D.  M. 

H.  M- 

H.  M. 

$ 

MERCURY 

Gr.  Elong.  19d. 

Inf  6 

Id.  llh. 

1 

10.  7.44 

6.55S 

4.  9.36 

4.54S 

13.  8N 

8.43 

23.51 

4 

10.18.14 

7.  0 

4.  7.24 

4.38 

13.57 

8.34 

23.32 

7 

10.29.34 

6.49 

4.  5.37 

4.  7 

14.54 

8.28 

23.15 

10 

11.11.53 

6.19 

4.  4.34 

3.25 

15.49 

8.24 

23.  1 

13 

11.25.22 

5.26 

4.  4.31 

2.35 

16.38 

.  8.25 

22.52 

16 

0.10.  9 

4.  8 

4.  5.33 

1.43 

17.14 

8.30 

22.47 

19 

0.26.18 

2.24 

4.  7.41 

0.52 

17.32 

8.39 

22.47 

22 

1.13.43 

0.18S 

4.10.51 

0.  6S 

17.26 

8.53 

22.50 

25 

2.  2.  7 

1.56N 

4.14.55 

0.34N 

16.55 

9.10 

22.57 

28 

2.21.  2 

4.  1 

4.19.43 

1.  5 

15.57 

9.30 

23.  7 

31 

3.  9.47 

5.38 

4.25.  2 

1.28 

14.34 

9.51 

23.17 

2 

VENUS. 

1 

5.25.39 

3.20N 

4.28.  1 

1.30N 

13.35N 

10.  3 

1.18 

7 

6.  5.22 

3.11 

5.  5.22 

1.28 

10.55 

10.31 

1.23 

13 

6.15.  3 

2.56 

5.12.42 

1.22 

8.  4 

10.58 

1.27 

19 

6.24.43 

2.37 

5.20.  3 

1.15 

5.  5 

11.25 

1.32 

25 

7.  4.21 

2.13 

5.27.22 

1.  4 

2.  2  ' 

11.52 

1.37 

$ 

MARS. 

1 

1.24.16 

0.12N 

2.23.19 

0.  9N 

23.27N 

5.31 

20.45 

7 

1.27.32 

0.18 

2.27.19 

0.14 

23  40 

5.48 

20.39 

13 

2.  0.45 

0.24 

3.  1.17 

0.19 

23.46 

6.  6 

20.34 

19 

2.  3.56 

0.30 

3.  5.11 

0.24 

23.45 

6.23 

20.28 

25 

2.  7.  6 

0.36 

3.  9.  3 

0.29 

23.38 

6.40 

20.23 

% 

JUPITER. 

1 

0.17.59 

1.18S 

0.29.39 

1.22S 

10.  5N 

1.52 

17.  5 

7 

0.18.32 

1.18 

0.29.57 

1.24 

10.10 

1.53 

16.43 

13 

0.19.  5 

1.18 

1.  0.  9 

1.25 

10.12 

1.54 

16.21 

19 

0.19.38 

1.18 

1.  0.13 

1.27 

10.12 

1.55 

15.59 

25 

0.20.11 

1.17 

1.  0.10 

1  28 

10.10 

1.54 

15.36 

k 

SATURN. 

1 

0.20.34 

2.  SOS 

0.26.40 

2.34S 

7.54N 

1.43 

16.55 

7 

0.20.47 

2.30 

0.26.43 

2.36 

7.53 

1.43 

16.32 

13 

0.20.59 

2.30 

0.26.42 

2.38 

7.51 

1.43 

16.  9 

19 

0.21.12 

2.30 

0.26.37 

2.39 

7.48 

1.43 

15.47 

25 

0.21.24 

2.30 

0.26.28 

2.41 

7.44 

1.42 

15.24 

¥ 

GEORGIAN. 

1 

9.  1.33 

0.15S 

8.29.39 

0.15S 

23.43S 

17  58 

9.12 

11 

9.  1.40 

0.15 

8.29.24 

0.15 

23.43 

17.57 

8.33 

21 

9.  1.47 

0.15 

8.29.14 

0.15 

23.43 

17.57 

7.55 

TABLE  LV< 


67 


Nautical  Almanac, 
V.  AUGUST  1821. 


0) 

<u 

£ 

0) 

£ 

z 

o 

s 

0) 

THE  MOON’S 

JZ 

o 

u- 

o 

Longitude. 

Latitude. 

cn 

CS 

Hi 

X 

Q 

Noon. 

Midnight. 

Noon. 

Midnight. 

S.  D.  M.  S. 

S-  D.  M.  S. 

D.  M.  S. 

D.  M.  S. 

w. 

1 

5.18.28.49 

5.24.32.24 

1.14.51  S 

1.46.44  S 

Th. 

2 

6.  0.33.19 

6.  6.32.  3 

2.17.13 

2.46.  1 

F. 

3 

6.12.29.  7 

6.18.25.  3 

3.12.53 

3.37.36 

Sa. 

4 

6.24.20.24 

7.  0.15.46 

3.59.56 

4.19.43 

Sun. 

5 

7.  6.11.44 

7.12.  8.55 

4.36.46 

4.50.53 

M. 

6 

7.18.  7.55 

7.24.  9.18 

5.  1.55 

5.  9.43 

Tu. 

7 

8.  0.13.38 

8.  6.21.29 

5.14.  8 

5.15.  1 

W. 

8 

8.12.33.19 

8.18.49.35 

5.12.17 

5.  5.48 

Th. 

9 

8.25.10.40 

9.  1.36.53 

4.55.30 

4.41.22 

F. 

10 

9.  8.  8.27 

9.14.45.29 

4.23.25 

4.  1.42 

Sa. 

11 

9.21.28.  0 

9.28.15.54 

3.36.22 

3.  7.39 

Sun. 

12 

10.  5.  9.  1 

10.12.  7.  1 

2.35.50 

2.  1.18 

M. 

13 

10.19.  9.30 

10.26:15.57 

1.24.33 

0.46.  7  S 

Tu. 

14 

11.  3.25.47 

11.10.38.23 

0.  6.38  S 

0.33.14  N 

W. 

15 

11.17.53.  4 

11.25.  9.  6 

1.12.45  N 

1.51.14 

Th. 

16 

0.  2.25.49 

0.  9.42.32 

2.27.58 

3.  2.17 

F. 

17 

0.16.58.37 

0.24.13.27 

3.33.37 

4.  1.26 

Sa. 

18 

1.  1.26.31 

1.  8.37.23 

4.25.17 

4.44.50 

Sun. 

19 

1.15.45.41 

1.22.51.  6 

4.59.50 

5.10.10 

M. 

20 

1.29.53.21 

.2.  6.52.15 

5.15.45 

5.16.37 

Tu. 

21 

2.13.47.41 

2.20.39.33 

5.12.51 

5.  4.38 

W. 

22 

2.27.27.49 

3.  4.12.27 

4.52.12 

4.35.49 

Th. 

23 

3.10.53.27 

3.17.30.50 

4.15.49 

3.52.32 

F 

24 

3.24.  4.38 

4.  0.34.54 

3.26.23 

2.57.48 

Sa. 

25 

4.  7.  1.42 

4.13.25.  6 

2.27.11 

1.54.58 

Sim. 

26 

4.19.45.10 

4.26.  2.  0 

1.21.36 

0.47.30  N 

M. 

27 

5.  2.15.44 

5.  8.26.28 

0.13.  7  N 

0.21.  8  S 

Tu. 

28 

5.14.34.24 

5.20.39.41 

0.54.53  S 

1.27.46 

W. 

29 

5.26.42.33 

6.  2.43.14 

1.59.26 

2.29.35 

Th. 

30 

6.  8.42.  3 

6.14.39.19 

2.57.55 

3.24.11 

F. 

31 

6.20.35.23 

6.26.30.38 

3.48.  9 

4.  9.37  1 

68 


TABLE  LV. 


Nautical  Almanac. 


AUGUST  1821.  VI, 


M 

<L> 

<u 

e 

o 

THE  MOON’S. 

1 

£ 

u 

u- 

<u 

Age. 

Passage 

Right  Ascension. 

Declination. 

o 

C3 

o 

CO 

Me  rid. 

JYoon. 

JWidnight. 

J\'oon. 

Midnight. 

Q 

Q 

D. 

H.  M. 

D.  M. 

D.  M. 

D.  M. 

D.  M. 

w. 

1 

5 

2.35 

168.55 

174.17 

3.25  N 

0.32  N 

Th. 

2 

6 

3.15 

179.36 

184.54 

2.19  S 

5.  8  S 

F. 

3 

7 

3.54 

190.13 

195.35 

7.54 

10.35 

Sa. 

4 

8 

4.35 

201.  2 

206.35 

13.10 

15.38 

Sun. 

5 

9 

5.18 

212.16 

218.  7 

17.57 

20.  6 

M. 

6 

10 

6.  4 

224.  9 

230.23 

22.  5 

23.50 

Tu. 

7 

11 

6.53 

236.50 

243.29 

25.20 

26.34 

W. 

8 

12 

7.46 

250.20 

257.22 

27.29 

28.  4 

Th. 

9 

13 

8.42 

264.33 

271.50 

28.18 

28.  9 

F. 

10 

14 

9.39 

279.10 

286.31 

27.36 

26.39 

Sa. 

11 

15 

10.36 

293.50 

301.  4 

25.19 

23.36 

Sun. 

12 

16 

11.31 

308.11 

315.11 

21.31 

19.  7 

M. 

13 

17 

12.23 

322.  3 

328.47 

16.26 

13.30 

Tu. 

14 

18 

13.14 

335.24 

341.55 

10,.  22 

7.  5 

\V. 

15 

19 

14.  3 

348.23 

354.49 

3.41  S 

0.14  S 

Th. 

16 

20 

14.53 

1.15 

7.43 

3.14  N 

6.39  N 

F. 

17 

21 

15.44 

14.15 

20.54 

9.58 

13.  9 

Sa. 

18 

22 

16.37 

27.41 

34.38 

16.  8 

18.53 

Sun. 

19 

23 

17.34 

41.45 

49.  1 

21.21 

23.30 

M. 

20 

24 

18.33 

56.27 

64.  2 

25.17 

26.40 

Tu. 

21 

25 

19.33 

71.43 

79.26 

27.39 

28.12 

W. 

22 

26 

20.32 

87.  8 

94.45 

28.19 

28.  0 

Th. 

23 

27 

21.29 

102.14 

109.33 

27.16 

26.  9 

F. 

24 

28 

22.21 

116.38 

123.29 

24.42 

22.56 

Sa. 

25 

29 

23.  9 

130.  6 

136.28 

20.54 

18.39 

Sun. 

26 

30 

23.54 

142.37 

148.34 

16.12 

13.36 

M. 

27 

1 

6 

154.20 

159.57 

10.53 

8.  5 

Tu. 

28 

2 

0.36 

165.27 

170.51 

5.14  N 

2.22  N 

W. 

29 

3 

1.17 

176.11 

181.30 

0.31  S 

3.22  S 

Th. 

30 

4 

1.57 

186.49 

<192.  9 

6.11 

8.55 

F. 

31 

5 

2.37  . 

197.33 

203.  2 

11.34 

14.  7 

1 

TABLE  LV 


69 


Nautical  Almanac . 


VII.  AUGUST  1821. 


Days  of  the  Week. 

Days  of  the  Month. 

THE  MOON’S 

I 

rlional 

ithm. 

| 

j 

Semidiameter. 

Iior.  Parallax. 

Propoi 

Logur 

Noon. 

Midnight 

Noon. 

Midnight 

M.  S. 

M.  S. 

M.  S. 

M.  S. 

Noon,  j 

Midnight 

W. 

1 

14.59 

14.55 

54.53 

54.41 

5158 

5174 

Th. 

2 

14.53 

14.50 

54.31 

54.22 

5187 

5199  ! 

F. 

3 

14.49 

14.47 

54.16 

54.12 

5207 

5213  l 

Sa. 

4 

14.47 

14.47 

54.11 

54.12 

5214 

5213 

Sun. 

5 

14.49 

14.50 

54.16 

54.22 

5207 

5199 

M. 

6 

14.53 

14.56 

54.31 

54.42 

5187 

5173 

Tu. 

7 

14.59 

15.  4 

54.55 

55.11 

5156 

5135 

W. 

8 

15.  8 

15.14 

55.29 

55.49 

5111 

5085 

Th. 

9 

15.20 

15.26 

56.10 

56.33 

5058 

5028 

F- 

10 

15.32 

15.39 

56.57 

57.21 

4998 

4967 

Sa. 

11 

15.46 

15.52 

57.45 

58.  8 

4937 

4908 

Sun. 

12 

15.58 

16.  4 

58.30 

58.51 

4881 

4855 

M. 

13 

16.  9 

16.13 

59.10 

59.26 

4832 

4812 

Tu. 

14 

16.17 

16.20 

59.40 

59.51 

4795 

4782 

W. 

15v 

16.22 

■16.23 

59.59 

60.  3 

4772 

4768 

Th. 

16 

16.24 

16.23 

60.  5 

60.  3 

4765 

4768 

F. 

17 

16.22 

16.20 

59.58 

59.51 

4774 

4782 

Sa. 

18 

16.17 

16.14 

59.42 

59.31 

4793 

4806 

Sun. 

19 

16.11 

16.  7 

59.18 

59.  4 

4822 

4839 

M. 

20 

16.  3 

15.59 

58.50 

58.35 

4856 

4875 

Tu. 

21 

15.55 

15.50 

58.19 

58.  3 

4895 

4915 

W. 

22. 

15.46 

15.42 

57.17 

57.31 

4935 

4955 

Th. 

23 

15.37 

15.33 

57.15 

56.59 

4975 

4995 

F. 

24 

15.29 

15.25 

56.44 

56.29 

5014 

5034 

Sa. 

25 

15.21 

15.17 

56.14 

55.59 

5053 

5072 

Sun. 

26 

15.13 

15.  9 

55.45 

55.31 

5090 

5108 

M. 

27 

15.  5 

1  15.  2 

55.18 

55.  5 

5125 

5143 

Tu. 

28 

14.59 

14.56 

54.53 

54.42 

5158 

5173 

W. 

29 

14.53 

14.50 

54.32 

54.23 

5186 

5198 

Th. 

30 

14.49 

14.47 

54.16 

54.10 

5207 

5215 

F. 

31 

14.46 

14.45 

54.  6 

54.  4 

5221 

5223 

DISTANCES  OF  MOON’S  CENTER  FROM  SUN,  AND  FROM  STARS  EAST  OF  HER. 
I  1  Noon.  j  Illh.  j  VIh.  j  lXh.  £  Midnight.  \  XVh.  i  XYlIlh. 


70 


TABLE  LV, 


Nautical  Almanac, 
AUGUST  1821. 


VII  f. 


! 


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IT) 


KCTQOO 
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CTHOOO 
CO  KO  ^ 


CN  00  ^ 
t'-  *0  ■'3* 


h-tOO'f 

CM 


OO'O'^'WO 
N  (O  V3  ^  W 


rt 

Q 


CN  CO  Tjt  V5 


VXD 


rt  S 


3 

CJ* 

< 

53 


bJ3 

<L> 

PU 


table  lv, 


Nautical  Almanac . 
IX.  AUGUST  1821. 


CO 

O  lO 

O  rH 

oo  rj<  at  co  co  to 

b-  to  to 

to  io 

rH  rtf 

CM  to  rtf  to 

tO  CM  rtf  - 

t  Cj 

1— < 

C5t  rH 

rtf  ot 

CONOO'^Nr}' 

tO  CM  CM 

X 

rtf  V3 

CM  rH 

rH  to  rtf  to  rH  rtf 

tO  rH 

tO  CM 

tO  tH  oo  >o  co  O 

Oi  oo  to 

d 

tO  CO 

to  to 

H  O  09  b-  tO  *0 

tO  rtf  CO 

CO 

b-  to 

o  cn 

at  rH  co  O  co  rH  CO 

CO  rtf  at 

lo  rH 

rtf  CM 

to  to  rj<  H  H  Tj<  H 

rtf  CM  rH 

• 

to  to 

d  rtf 

00  to  to  O  N.  K  rH 

d  rH  rH 

co  co 

rH 

tO  CO  CM  CO  rtf  rH 

CM  CO  rtf 

X 

. 

rtf  d 

00  rtf 

co  co  o  bl  d  cm  o 

rH  Ot  bl 

p 

to  rtf 

to  to 

h  o  at  b.  to  to  rj« 

to  rtf  co 

CO 

to  to 

Ot  O  O  ©  rH  at  rtf 

cm  at  to 

rH 

CM  CO 

to  co  to  co  *o  co  co 

rtf  CM  to 

•H 

CM  Ot 

bl  at 

Ot  to  ^  to  H  O  CM 

to  d  d 

X 

CM  rH 

to  rtf 

CO  rH  CM  to  CO 

to 

to  cm 

d  to 

00  to  CM  Ot  to  co  rH 

CM  rH  Ot 

d 

to  rtf 

to  to 

rH-O  Ot  b-  to  to  ^ 

rH  rH 

CO  to  CO 

CO 

at  co  cm 

co  to 

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CO  00  CM 

■c 

rtf  CM  cm 

rH 

CM  rtf  to 

rtf  CO  CO 

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M. 

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d  d 

rtf  CO 

H  to  CO  CO  to  CO  ^ 

CM  to  rtf  rtf  to  CM 

to  at  oo 

CM  GM  CO 

§ 

00  rtf  o 

rH  bl 

d  d  co  o  k  d  co 

rtf  CM  O 

d 

to  rtf  co 

b-  to 

cm  o  at  oo  to  to  rtf 

tO  to  rtf 

. 

CM  rH  O 

rtf 

rH  CM  at  rtf  co  co 
•  CM  CO  rtf  rtf  rH  to 

CO  CM 

CZ3 

co  .  co 

l  1— ( 

.  to  rH 

JC 

M. 

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to  rtf 

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rtf 

CM  GM  b-  at  00  rH 

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s 

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CM  CO 

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to  rtf  co 

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to  b,  K 

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t— I 

to  CO  Ot 

.  bl  d 

,  b~  00  rH-  00  00  CM 

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rH 

crt  ^ 

CO  TH  rH  rH  co  rH 

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to  H  K 

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1  CO  O  bl  rtf  rH  Ot 

,  00  CO  rtf 

P 

to  *o  CO 

to  to 

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to  rtf  CO 

c 

n 

ON00 

oo  at  o 

K  00  Ot  O  H  CM  CO 

OlOHH 

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H  H  H 

rH  rH  CM 

rH  rH  rH  CM  CM  CM  CM 

CM  CO  CO 

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CO 

05 

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to 

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c 

es 

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73 


TABLE  LV 


Nautical  Almanac. 


AUGUST  1821.  X. 


CO* 

HOiKMiliO'# 

00  rH  rtf  oo  © 

co  O,  K  K  CM 

-C 

CO  CN  CN  -q1  W  rH  ri 

.  CM  rH  CM 

VO  VO  CM  rH 

55 

•OVicinNHCO 

■tf  -tf  CO  oo  CM 

CO  CM  CO  rH  rH 

<< 

cm  H  H  rH  CM 

CM  CM  CO 

CM  CM  VO 

55 

p 

O)  O  H  CN  CC  *o 

vo  N.  Oo  CM  Vo 

CM  co  ©  tf  00 

^okooo)Oh 

CM  CO  -tf  CO  is. 

tf  Vo  is.  OO  Oo 

CO* 

i-COCOOlOOrt 

(MOiK<OK 

CO  CM  H  h.  CO 

-d 

CIN»OH  CO  tf 

VO  CO  CO  -tf  CO 

CM  CM  rH  vo 

a 

q 

WcoKO'OKa 

tf  co  VO  tf  tf 

CO  Oi  K  CO  H 

2 

> 

»CJ>Ori<cf 

V)  VO  CO  CM 

tf  rH  rH  CO 

55 

p 

00  O)  a  o’  rH  cJ  rf< 

CO  VO  CO  O  CO 

O  tf  CO  CM  is! 

o 

^^^OCOO^OH 

rH  rH 

CM  CO  tf  ©  is. 

tf  vo  CO  00  OO 

j 

X) 

CO* 

CO  O  K  *0  *0  *C  O 

VO  O  V)  CO  O 

CM  CO  h  Oo  O 

H  H  CO  »0  H  CO 

tf  rH  i-h  rH 

tf  CO  CM  vo 

-H 

> 

5> 

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VO  CO  CO  O  b- 

CM  CO  rH  VO  CM 

CO  tf  co  CM  CM  CM  CO 

CM  CM  CO  tf 

CO  CO  tf  rH 

55 

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lOKCOOOwO) 

CM  tf  CO  Oo  rH 

oi  o'  co  o'  ‘n 

p 

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CM  co  tf  VO 

CO  vo  co  CO  oo 

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rH  H 

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CO 

cm  tf  ©  co  hh  cm  cm 

N.  CO  O  O  vo 

O  O  OO  CO  CO 
tf  rH  vo  vo  vo 

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p 

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TABLE  LV, 


73 


Nautical  Almanac . 


XI.  AUGUST  1821. 


aj 

T?  O 

h.  cn 

to  to 

■y  yen 

cn  to 

»0  Tf 

co  co 

rH  hP 

CN  *0  CN 

CN  rH 

cn  co 

CN  y 

CN  d 

CN  CN  -H 

O  CN 

X! 

IS 

H}<  to 

cn  y  to 

y  co 

* 

Tp  tO 

CN  »0 

to  cn 

d  co  to 

r-f 

p 

to  hr 

to  h. 

to  h- 

to  to  hr 

^  *0 

cc 

to  cO 

o  >o 

rH  hr 

CO  00  to 

co  y 

JO 

•  ^ 

y  CN 

CN  co 

CO  CN  CO 

cn  y 

zz 

*o  d 

o  y 

CO  CN 

rH  co  co 

oo  d 

•—1 

rO 

CO  CO 

to 

rH 

y  rH 

.X 

• 

CN  y 

ss 

to  oo 

00  CN  to 

O  rH 

tO  h- 

to  h. 

Tp  tO  h- 

y  »o 

v» 

co  oo 

co  00 

y  o 

CN  CN  00 

tO  to 

to 

CN  CN 

CO  to  CN 

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— 

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y  y 

o  co  d 

cn 

V) 

rH  CN 

CN  CO 

.  ^  tv2 

'  ^ 

X 

1-1  CN 

cn  cn 

CO  d 

h.  O  CO 

d 

- 1 

VO  h- 

to  h- 

to  h- 

y  to  hr 

y 

02 

T?  O 

o  cn  cn 

to  rf 

rH  y  rH  rH 

■i* 

co  co 

CN  y  CO 

CN  »0 

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to 

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to  to 

cn  -y  h.  00 

1  hr 

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*?; 

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h  y  »o  to 

§ 

• 

Cn  rH 

hoco 

rH  Tf 

tO  CO  rH  Tp 

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2 

to  h- 

to  h.  00 

to  h- 

y  to  h.  oo 

y 

CO 

00  rH 

cn  cn  oo 

rH 

o  >o  cn  y 

o 

to  CN 

GN  y 

•  GN 

y  rH 

co 

£Z 

„ 

to  cn 

to  to  cn 

00  y  00  rH 

X 

rH 

to  rH 

. 

CO  rH  CN 

1 

2 

co  cn 

d  cn  CN 

•  CO 

CONOCO 

y  to  h.  oo 

• 

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to  to  00 

hr 

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CO 

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CO  to  H  to 

hr 

CN  rH 

CN  to  to 

i  to 

to  co  h  y 

’  CN 

j= 

s 

hi  © 

CN 

cn  hi  th 

CO  y 

,  d 

to 

y  CJ  rH  tO 

■S3 

to  h. 

cn  to  cn 

«  cn 

O  CO  hr  O 

p 

to  to 

to  to  h. 

to 

y  to  to  oo 

a5 

CO  00  to 

hr  y  rH 

cn  to 

to  to  00  co  . 

o  y 

y  rH  y 

y  to 

•  y 

'  to  CN 

2* 

S 

«M 

y  rH  CN 

CN  00  CN 

,  rH  00 

co  co  fN  cn 

,  rH  CO 

CO  CO 

CN 

’  CN  CN 

CO  CN  CN 

tO 

Q 

y  *o  hr 

H  rj<h 

.  CO  rH 

00  CN  to  00 

i  co  co 

to  to  K 

to  to  h- 

to  00 

CO  to  to  h- 

y  to 

C/3 

>* 

^  O  K 

h-  co  cn 

Cn  O  rH 

rH  cn  co  y 

OHH 

Q 

H  H  H 

rH  rH  rH 

rH  CN  CN 

CN  CN  CN  CN 

CO  co  * 

Jj 

x? 

CO 

ctt 

cn 

V 

jy 

"5 

3 

2 

'« 

c« 

bo 

eg 

V 

c 

o 

rJl 

5 

C* 

<D 

u 

c« 

5 

< 

a 

e 

o 

u* 

Du 

a 

< 

3 

<U 

JO 

u 

*10 


74 


TABLE  LVI, 


Second  Differences. 


Jfio 

urs 

&  Min- 

1' 

2' 

3' 

4'  1 

5' 

6' 

7' 

8'  | 

9' 

10' 

h 

m 

h 

m 

!  o 

0 

12 

0 

0".0 

0".0 

0".0 

0".0 

0".0 

0".0 

0".0 

0".00".0 

0".0 

0".0 

0 

10 

11 

50 

0.4 

0.8 

1.2 

1.6 

2.0 

2.4 

2.9 

3.3! 

3.7 

4.1 

4.5 

0 

20 

11 

40 

0.8 

1.6 

2.4 

3.2 

4.1 

4.9 

5.7 

6.5j 

7.3 

8.1 

8.9 

0 

30 

11 

30 

1.2 

2.4 

3.6 

4.8 

6.0 

7.2 

8.4 

9.6T0.8 

12.0 

13.2 

0 

40 

11 

20 

1.6 

3.1 

4.7 

6.3 

7.9 

9.4 

11.0 

12. 614. 9 

15.7 

17.3 

0 

50 

11 

10 

1.9 

3.9 

5.8 

7.8 

9.7 

11.6 

13.6 

15.5(17.4 

19.4 

21.4 

i 

0 

11 

0 

2.3 

4.6 

6.9 

9.2 

11.5 

13.8 

16.0 

18. 31 

20.6 

22.9 

25.2 

i 

10 

10 

50 

2.6 

5.3 

7.9 

10.5 

13.2 

15.8 

18.4 

21.123.7 

26.3 

29.0 

i 

20 

10  40 

3.0 

5.9 

8.9 

11.9 

14.8 

17.8 

20.7 

23.7i26.7 

29.6 

32.6 

l 

30 

10 

30 

3.3 

6.6 

9.8 

13.1 

16.4 

19.7 

23.0 

26.3| 

29.5 

32.8 

36.1 

i 

40 

10 

20 

3.6 

7.2 

10.8 

14.4 

17.9 

21.5 

25.1 

28.7'32.3 

35.9 

39.5 

i 

50 

10 

10 

3.9 

7.8 

11.6 

15.5 

19.4 

23.3 

27.2 

31.0|34.9 

38.8 

42.7 

2 

0 

10 

0 

4.2 

8.3 

12.5 

16.7 

20.8 

25.0 

29.2 

33.3 

137.5 

41.7 

45.8 

2 

10 

9 

50 

4.4 

8.9 

13.3 

17.8 

22.2 

26.6 

31.1 

35.540.0 

44.4 

48.8 

2 

20 

9 

40 

4.7 

9.4 

14.1 

18.8 

23.5 

28.2 

32.9 

37. 6|42. 3 

47.0 

51.7 

2 

30 

9 

30 

4.9 

9.9 

14.8 

19.8 

24.7 

29.7 

34.6 

39.6J44.5 

49.5 

54.4 

2 

40 

9 

20 

5.2 

10.4 

15.6 

20.7 

25.9 

31.1 

36.3 

41.5;46.7 

51.9 

57.0 

2 

50 

9 

10 

5.4 

10.8 

16.2 

21.6 

27.1 

32.5 

37.9 

43.3|48 .7 

54.1 

59.5 

3 

0 

9 

0 

5.6 

11.3 

16.9 

22.5 

28.1 

33.8 

39.4 

45.0:50.6 

56.3 

61.9 

3 

10 

8 

50 

5.8 

11.7 

17.5 

23.3 

29.1 

35.0 

40.8 

46.6,52.4 

58.3 

64.1 

3 

20 

8 

40 

6.0 

12.0 

18.1 

24.1 

30.1 

36.1 

42.1 

48.1  54.2 
1 

60.2 

66.2 

.3 

30 

8 

30 

6.2 

12.4 

18.6 

24.8 

31.0 

37.2 

43.4 

49.6;55.8 

62.0 

68.2 

3 

40 

8 

20 

6.4 

12.7 

19.1 

25.5 

31.8 

38.2 

44.6 

50.957.3 

63.7 

70.0 

3 

50 

8 

10 

6.5 

13.0 

19.6 

26.1 

32.6 

39.1 

45.7 

52.258.7 

65.2 

71.7 

4 

0 

8 

0 

6.7 

13.3 

20.0 

26.7 

33.3 

40.0 

46.7 

53.3 

160.0 

66.7 

73.3 

4 

10 

7 

50 

6.8 

13.6 

20.4 

27.2 

34.0 

40.8 

47.6 

54.461.2 

68.0 

74.8 

4 

20 

7 

40 

6.9 

13.8 

20.8 

27.7 

34.6 

41.5 

48.4 

55.4,62.3 

69.2 

76.1 

4 

30 

7 

30 

7.0 

14.1 

21.1 

28.1 

35.2 

42.2 

49.2 

56.263.3 

70*3 

77.3 

4 

40 

7 

20 

7.1 

14.3 

21.4 

28.5 

35.6 

42.8 

49.9 

57.0164.2 

71.3 

78.4 

4 

50 

7 

10 

7.2 

14.4 

21.6 

28.9 

36.1 

43.3 

50.5 

57-7 

64.9 

72.2 

79.4 

5 

0 

7 

0 

7.3 

14.6 

21.9 

29.2 

36.5 

43.8 

51.0 

58.3 

165.6 

72.9 

80.2 

5 

10 

6 

50 

7-4 

14.7 

22.1 

29.4 

36.8 

44.1 

51.5 

58.8 

66.2 

73.6 

80.9 

5 

20 

6 

40 

7.4 

14.8 

22.2 

29.6 

37.0 

44.4 

51.9 

59.3 

66.7 

74.1 

81.5 

5 

30  i 

6 

30 

7.4 

14.9 

22.3 

29.8 

37.2 

44.7 

52.1 

59.6 

67.0 

74.5 

81.9 

5 

40 

6 

20 

7.5 

15.0 

22.4 

29.9 

37.4 

44.9 

52.3 

59.8 

67.3 

74.8 

82.2 

5 

50 

6 

10 

7.5 

15.0 

22.5 

30.0 

37.5 

45.0 

52.5 

60.0 

67.4 

74.9 

82.4 

]  6 

0 

6 

0 

7.5 

15.0 

22.5|30.0 

37.5 

45.0 

52.5 

60.0 

67.5175.0 

82.5 

TABLE  LVI 


75 


Second  Differences, 


|  Hours  &  Min 

.  10' 

'20' 

'30' 

'40' 

'50" 

1" 

2" 

3V 

4" 

5" 

6" 

•  7" 

9" 

h  m 

h  m 

n 

// 

// 

// 

v  // 

// 

// 

// 

0  0 

12  0 

0.( 

)0.C 

)0.( 

30. C 

o'.o 

O.C 

)0.C 

)  0-C 

O.C 

3  O.C 

30. C 

)  0.( 

3  O.C 

1  o'.o 

0  10 

11  50 

0.1 

0.1 

0.5 

20.3 

0.3 

o.c 

)  o.c 

)0.c 

O.C 

30. C 

30. C 

3  0.( 

3  0.1 

L  0.1 

0  20 

11  40 

0.1 

10.: 

0.^ 

1 0.5 

0.7 

O.C 

)0.C 

)  o.c 

0.1 

0.1 

L  0.1 

.  0.] 

L  0.1 

.  0.1 

0  30 

11  30 

o.s 

10.4 

O.t 

50.8 

1.0 

o.c 

)0.C 

)0.1 

0.1 

.  0.1 

L  0.1 

0.1 

L  O.S 

!  0.2 

0  40 

11  20 

o.: 

0.5 

O.C 

3  l.C 

1.3 

o.c 

30.1 

L  0.1 

0.1 

.0.1 

L  0.2 

!  O.S 

1  O.S 

!  0.2 

0  50 

11  10 

o.: 

10. c 

l.C 

1.3 

1.6 

o.c 

)  0.1 

L  0.1 

0.1 

.O.S 

10. S 

!  O.S 

1  0.3 

1  0.3 

1  0 

11  0 

0.4 

o.s 

1.1 

1.5 

1.9 

o.c 

>0.1 

.0.1 

o.s 

!  O.S 

10.2 

1*0.3 

5  0.3 

!  0.3 

1  10 

10  50 

0.4 

0.9 

1.: 

1.8 

2.2 

o.c 

>0.1 

.0.1 

o.s 

!  O.S 

10.3 

:  0.' 

>  0.4 

;  0.4 

1  20 

10  40 

0.5 

1.0 

1.5 

2.0 

2.5 

o.c 

10.1 

0.1 

o.s 

!  O.S 

!  0.3 

;  O.c 

3  0.4 

<  0.4 

1  30 

10  30 

0.5 

1.1 

1.6 

2.2 

2.7 

0.1 

0.1 

0.2 

0.2 

!0.: 

l  0.3 

0.4 

L  0.4 

•  0.5 

1  40 

10  20 

0.6 

1.2 

1.8 

2.4 

3.0 

0.1 

0.1 

0.2 

0.2 

;  0.3 

;  0.4 

0.4 

L  0.5 

;  0.5 

1  50 

10  10 

0.6 

1.3 

1.9 

2.6 

3.2 

0.1 

0.1 

0.2 

0.3 

0.3 

;  0.4 

0.5 

>  0.5 

0.6 

2  0 

10  0 

0.7 

1.4 

2.1 

2.8 

3.5 

0.1 

0.1 

0.2 

0.3 

0.3 

0.4 

0.5 

0.6 

0.6 

2  10 

9  50 

0.7 

1.5 

2.2 

3.0 

3.7 

0.1 

0.1 

0.2 

0.3 

0.4 

0.4 

0.5 

0.6 

0.7 

2  20 

9  40 

0.8 

1.6 

2.3 

3.1 

3.9 

0.1 

0.2 

0.2 

0.3 

0.4 

0.5 

0.5 

0.6 

0.7 

2  30 

9  30 

0.8 

1.6 

2.5 

3.3 

4.1 

0.1 

0.2 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.7 

2  40 

9  20 

0.9 

1.7 

2.6 

3.5 

4.3 

0.1 

0.2 

0.3 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

2  50 

9  10 

0.9 

1.8 

2.7 

3.6 

4.5 

0.1 

0.2 

0.3 

0.4 

0.5 

0.5 

0.6 

0.7 

0.8 

3  0 

9  0 

0.9 

1.9 

2.8 

3.8 

4.7 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.7 

0.8 

3  10 

8  50 

1.0 

1.9 

2.9 

3.9 

4.9 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

3  20 

8  40 

1.0 

2.0 

3.0 

4.0 

5.0 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

3  30 

8  30 

1.0 

2.1 

3.1 

4.1 

5.2 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

3  40 

8  20 

1.1 

2.1 

3.2 

4.2 

5.3 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

1.0 

3  50 

8  10 

1.1 

2.2 

3.3 

4.3 

5.4 

0.1 

0.2 

0.3 

0.4 

0.5 

0.7 

0.8 

0.9 

1.0 

4  0 

8  0 

1.1 

2.2 

3.3 

4.4 

5.6 

0.1 

0.2 

0.3 

0.4 

0.6 

0.7 

0.8 

0.9 

1.0 

4  10 

7  50 

1.1 

2.3i 

3.4 

4.5. 

5.7 

0.1 

0.2 

0.3 

0.5 

0.6 

0.7 

0.8 

0.9 

1.0 

4  20 

7  40  : 

1.2 

2.3; 

3.5 

4.6 ; 

5.8 

0.1 

0.2 

0.3 

0.5 

0.6 

0.7 

0.8 

0.9 

1.0 

4  30 

7  30  : 

1.2 

2  3: 

3.5 

4.7: 

5.9 

0.1 

3.2 

0.4  < 

3.5 

0.6 

0.7 

0.8 

0.9 

1.1 

4  40 

7  20  : 

1.2 

2.4L 

3.6 

4.8: 

5.9 

0.1 

0.2 1 

0.4  ( 

3.5 

3.6i 

0.7 

0.8 

1.0 

1.1 

4  50 

7  io  : 

1.2: 

2.4L 

3.6 

4.8  1 

5.0 

0.1 

0.2 1 

0.4  ( 

3.5< 

3.6  < 

3.7 

0.8 

1.0 

1.1 

5  0 

7  0  ] 

L.2  ‘ 

2.4  C 

3.6  ■ 

4.9  ( 

5.1 

0.1 

3.21 

3.4  ( 

3.5  ( 

3.6  ( 

3.7 

0.9 

1.(3 

1.1 

5  10 

6  50  1 

l.2i 

2.5  C 

3.7  ■ 

4.9  ( 

5.1 

0.1  ( 

3.2  ( 

3.4  ( 

3.5  ( 

3.6  ( 

3.7 

0.9 

1.0 

1.1 

5  20 

6  40  ] 

1.2  ‘ 

2.5 c 

3.7  • 

4.9  ( 

5.1 

0.1  ( 

3.2  ( 

3.4  ( 

3.5  ( 

3.6  ( 

3.7 

0.9 

1.0 

1.1 

5  30 

6  30  1 

L.2 S 

2.5  3 

3.7  . 

5.0  ( 

5.2 

0.1  ( 

).2( 

3.4  C 

3.5  C 

3.6  C 

3.7 

0.9 

1.0 

1.1 

5  40 

6  20  1 

L.2  S 

2.5: 

3.7. 

5.06 

5.2 

0.1  ( 

3.2( 

3.4  C 

3.5  C 

3.6  C 

3.7 

0.9 

1.0 

1.1 

5  50 

6  10  }1 

.25 

2.53 

3.7  : 

5.0  6 

5.2 

0.1  ( 

).2( 

3.4  C 

3.5  C 

3.6  C 

3.7 

0.9 

1.0 

1.1 

6  0 

6  0  (1 

..3]5 

2.63 

3.8  ; 

5.06 

5.3 

0.1  ( 

).2( 

3.4  C 

3.5  C 

3.6  C 

3.7 

0.9 

1.0 

1.1 

70 


TABLE  LVIT, 


Logistical  Logarithms. 


0/ 

1' 

2' 

3' 

4' 

5 

6' 

7' 

0 

60 

120 

180 

240 

300 

360 

420 

0" 

00000 

17782 

14771 

13010 

11761 

10792 

10000 

9331 

1 

35563 

17710 

14735 

12986 

11743 

10777 

9988 

9320 

2 

32553 

1 7639 

14699 

12962 

11725 

1 0763 

9976 

9310 

3 

30792 

17570 

14664 

12939 

11707 

10749 

9964 

9300 

4 

29542 

17501 

14629 

12915 

11689 

10734 

9952 

9289 

5 

28573 

17434 

14594 

12891 

11671 

10720 

9940 

9279 

6 

27782 

17368 

14559 

12868 

11654 

10706 

9928 

9269 

7 

27112 

17302 

14525 

12845 

11636 

10692 

9916 

9259 

8 

26532 

17238 

14491 

12821 

11619 

10678 

9905 

9249 

9 

26021 

17175 

14457 

12798 

11601 

10663 

9893 

9238 

10 

25563 

17112 

14424 

12775 

11584 

10649 

9881 

9228 

11 

25149 

17050 

14390 

12753 

11566 

10635 

9869 

9218 

12 

24771 

16990 

14357 

12730 

11549 

10621 

9858 

9208 

13 

24424 

16930 

14325 

12707 

11532 

10608 

9846 

9198 

14 

24102 

16871 

14292 

12685 

11515 

10594 

9834 

9188 

15 

23802 

16812 

14260 

12663 

11498 

10580 

9823 

9178 

16 

23522 

16755 

14228 

12640 

11481 

10566 

9811 

9168 

17 

23259 

16698 

14196 

12618 

11464 

10552 

9800 

9158 

18 

23010 

16642 

14165 

12596 

11447 

10539 

9788 

9148 

19 

22775 

16587 

14133 

12574 

11430 

10525 

9777 

9138 

20 

22553 

16532 

14102 

12553 

11413 

10512 

9765 

9128 

21 

22341 

16478 

14071 

12531 

11397 

10498 

9754 

9119 

22 

22139 

16425 

14040 

12510 

11380 

10484 

9742 

9109 

23 

21946 

16372 

14010 

12488 

11363 

10471 

9731 

9099 

24 

21761 

16320 

13979 

12467 

11347 

10458 

9720 

9089 

25 

21584 

16269 

13949 

12445 

11331 

10444 

9708 

9079 

26 

21413 

16218 

13919 

12424 

11314 

10431 

9697 

9070 

27 

21249 

16168 

13890 

12403 

11298 

10418 

9686 

9060 

28 

21091 

16118 

13860 

12382 

11282 

10404 

9675 

9050 

29 

20939 

16069 

13831 

12362 

11266 

10391 

9664 

9041 

30 

20792 

16021 

13802 

12341 

11249 

10378 

9652 

9031 

TABLE  LVII 


// 


Logistical  Logarithms. 


0' 

1' 

2' 

3' 

1  4' 

5' 

6' 

7' 

0 

60 

120 

180 

240 

300 

360 

420 

30" 

20792 

16021 

13802 

12341 

i  11249 

10378 

9652 

9031 

31 

20649 

15973 

13773 

12320 

11233 

10365 

9641 

9021 

32 

20512 

15925 

13745 

12300 

11217 

10352 

9630 

9012 

33 

20378 

15878 

13716 

12279 

11201 

10339 

9619 

9002 

34 

20248 

15832 

13688 

12259 

11186 

10326 

9608 

8992 

35 

20122 

15786 

13660 

12239 

11170 

10313 

9597 

8983 

36 

20000 

15740 

13632 

12218 

11154 

10300 

9586 

8973 

37 

19881 

15695 

13604 

12198 

!  11138 

10287 

9575 

8964 

38 

19765 

15651 

13576 

12178 

11123 

10274 

9564 

8954 

39 

19652 

15607 

13549 

12159  | 

11107 

10261 

9553 

8945 

40 

19542 

15563 

13522 

12139 

11091 

10248 

9542 

8935 

41 

19435 

15520 

13495 

12119 

11076 

10235 

9532 

8926 

42 

19331 

15477 

13468 

12099 

11061 

10223 

9521 

8917 

43 

19228 

15435 

13441 

12080 

11045 

10210 

9510 

8907 

44 

19128 

15393 

13415 

12061 

11030 

10197 

9499 

8898 

45 

19031 

15351 

13388 

12041 

11015 

10185 

9488 

8888 

46 

18935 

15310 

13362 

12022  ' 

10999 

10172 

9478 

8879 

47 

18842 

15269 

13336 

12003 

10984 

10160. 

9467 

8870 

48 

18751 

15229 

13310 

11984  1 

10969 

10147 

9456 

8861 

49 

18661 

15189 

13284 

11965 

10954 

10135 

9446 

8851 

50 

18573 

15149 

13259 

11946 

10939 

10122 

9435 

8842 

51 

18487 

15110 

13233 

11927  ! 

10924 

10110 

9425 

8833 

52 

18403 

15071 

13208 

11908 

10909 

10098 

9414 

8824 

53 

18320 

15032 

13183 

11889 

10894 

10085 

9404 

8814 

54 

18239 

14994 

13158 

11871  1 

10880 

10073 

9393 

8805 

55 

18159 

14956 

13133 

11852 

10865 

10061 

9383 

8796 

56 

18081 

14918 

13108 

11834 

10850 

10049 

9372 

8787 

57 

18004 

14881 

13083 

11816 

10835 

10036 

9362 

8778 

58 

17929 

14844 

13059 

11797 

10821 

10024 

9351 

8769 

59 

17855 

14808 

13034 

11779 

10806 

10012 

9341 

8760 

60 

17782 

14771 

13010 

11761 

10792 

10000 

9331 

8751 

78 


TABLE  LV1I 


Logistical  Logarithms. 


8' 

9' 

10' 

11' 

12' 

13' 

14' 

15' 

1  I6'  1 

480 

540 

600 

660 

720 

780 

840 

900 

960 

0" 

8751 

8239 

7782 

7368 

6990 

6642 

6320 

6021 

5740 

1 

8742 

8231 

7774 

7361 

6984 

6637 

6315 

6016 

5736 

2 

8733 

8223 

7767 

7354 

6978 

6631 

6310 

6011 

5731 

3 

8724 

8215 

7760 

7348 

6972 

6625 

6305 

6006 

5727 

4 

8715 

8207 

7753 

7341 

6966 

6620 

6300 

6001 

5722 

5 

8706 

8199 

7745 

7335 

6960 

6614 

6294 

5997 

5718 

6 

8697 

8191 

7738 

7328 

6954 

6609 

6289 

5992 

5713 

7 

8688 

8183 

7731 

7322 

6948 

6603 

6284 

5987 

5709 

8 

8679 

8175 

7724 

7315 

6942 

6598 

6279 

5982 

5704 

9 

8670 

8167 

7717 

7303 

6936 

6592 

6274 

5977 

5700 

10 

8661 

8159 

7710 

7302 

6930 

6587 

6269 

5973 

5695 

11 

8652 

8152 

7703 

7295 

6924 

6581 

6264 

5968 

5691 

12 

8643 

8144 

7696 

7289 

6918 

6576 

6259 

5963 

5686 

13 

8635 

8136 

7688 

7283 

6912 

6570 

6254 

5958 

5682 

14 

8626 

8128 

7681 

7276 

6906 

6565 

6248 

5954 

5677 

15 

8617 

8120 

7674 

7270 

6900 

6559 

6243 

5949 

5673 

16 

8608 

8112 

7667 

7264 

6894 

6554 

6238 

5944 

5669 

17 

8599 

8104 

7660 

7257 

6888 

6548 

6233 

5939 

5664 

18 

8591 

8097 

7653 

7251 

6882 

6543 

6228 

5935 

5660 

19 

8582 

8089 

76 46 

7244 

6877 

6538 

6223 

5930 

5655 

20 

8573 

8081 

7639 

7238 

6871 

6532 

6218 

5925 

5651 

21 

8565 

8073 

7632 

7232 

6865 

6527 

6213  . 

.5920 

5646 

22 

8556 

8066 

7625 

7225 

6859 

6521 

6208 

5916 

5642 

23 

8547 

8058 

7618 

7219 

6853 

6516 

6203 

5911 

5637 

24 

8539 

8050 

7611 

7212 

6847 

6510 

6198 

5906 

5633 

25 

8530 

8043 

7604 

7206 

6841 

6505 

6193 

5902 

5629 

26 

8522 

8035 

7597 

7200 

6836 

6500 

6188 

5897 

5624 

27 

8513 

8027 

7590 

7193 

6830 

6494 

6183 

5892 

5620 

28 

8504 

8020 

7583 

7187 

6824 

6489 

6178 

5888 

5615 

29 

8496 

8012 

7577 

7181 

6818 

6484 

6173 

5883 

5611 

30 

8487 

8004 

7570 

7175 

6812 

6478 

6168 

5878 

5607  1 

TABLE  LVII 


79 


Logistical  Logarithms. 


8' 

9' 

10' 

11' 

12' 

13' 

14' 

15' 

16' 

480 

540 

600 

660 

720 

780 

840 

900 

960 

30" 

8487 

8004 

7570 

7175 

6812 

6478 

6168 

5878 

5607 

31 

8479 

7997 

7563 

7168 

6807 

6473 

6163 

5874 

5602 

32 

8470 

7989 

7556 

7162 

6801 

6467 

6158 

5869 

5598 

33 

8462 

7981 

7549 

7156 

6795 

6462 

6153 

5864 

5594 

34 

8453 

7974 

7542 

7149  ! 

6789 

6457 

6148 

5860 

5589 

35 

8445 

7966 

7535 

7143  | 

6784 

6451 

6143 

5855 

5585 

36 

8437 

7959 

7528 

7137  i 

6778 

6446 

6138 

5850 

5580 

37 

8428 

7951 

7522 

7131 

6772 

6441 

6133 

5846 

5576 

38 

8420 

7944 

7515 

7124  | 

6766 

6435 

6128 

5841 

5572 

39 

8411 

7936 

7508 

7118  I 

6761 

6430 

6123 

5836 

5567 

40 

8403 

7929 

7501 

7112  ! 

i 

6755 

6425 

6118 

5832 

5563 

41 

8395 

7921 

7494 

7106  j 

6749 

6420 

6113 

5827 

5559 

42 

8386 

7914 

7488 

7100  I 

6743 

6414 

6108 

5823 

5554 

43 

8378 

7906 

7481 

7093 

6738 

6409 

6103 

5818 

5550 

,  44 

8370 

7899 

7474 

7087 

6732 

6404 

6099 

5813 

5546 

45 

8361 

7891 

7467 

7081 

6726 

6598 

6094 

5809 

5541 

46 

8353 

7884 

7461 

7075  ; 

6721 

6393 

6089 

5804 

5537 

4  7 

8345 

7877 

7454 

7069 

6715 

6388 

6084 

5800 

5533 

48 

8337 

7869 

7447 

7063  | 

6709 

6383 

6079 

5795 

5528 

49 

8328 

7862 

7441 

7057 

6704 

6377 

6074 

5790 

5524 

50 

8320 

7855 

7434 

7050 

6698 

6372 

6069 

5786 

5520 

51 

8312 

7847 

7427 

7044 

6692 

6367 

6064 

5781 

5516 

52 

8304 

7840 

7421 

7038 

6687 

6362 

6059 

5777 

5511 

53 : 

8296 

7832 

7414 

7032 

6681 

6357 

6055 

5772 

5507 

54  i 

8288 

7825 

7407 

7026 

6676 

6351 

6050 

5768 

5503 

55 

8279 

7818 

7401 

7020 

6670 

6346 

6045 

5763 

5498 

56 

8271 

7811 

7394 

7014 

6664 

6341 

6040 

5758 

5494 

57 

8263 

7803 

7387 

7008 

6659 

6336 

6035 

5754 

5490 

58 

8255 

7796 

7381 

7002 

6653 

6331 

6030 

5749 

5486 

59 

8247 

7789 

7374 

6996 

6648 

6325 

6025 

5745 

5481 

60 

8239 

7782 

7368 

6990 

6642 

6320 

6021. 

5740 

5477 

80 


TABLE  LVII 


Logistical  Logarithms. 


17' 

18' 

19' 

20' 

21' 

22' 

23' 

24' 

|  25' 

1020 

1080 

1140 

1200 

1260 

1320 

1380 

1440 

1  1500 

0" 

5477 

5229 

4994 

4771 

4559 

4357 

4164 

3979 

1  3802 

1 

5473 

5225 

4990 

4768 

4556 

4354 

4161 

3976 

3799 

.  2 

5469 

5221 

4986 

4764 

4552 

4351 

4158 

3973 

3796 

3 

5464 

5217 

4983 

4760 

4549 

4347 

4155 

3970 

3793 

4 

5460 

5213 

4979 

4757 

4546 

4344 

4152 

3967 

3791 

5 

5456 

5209 

4975 

4753 

4542 

4341 

4149 

3964 

3788 

6 

5452 

5205 

4971 

4750 

4539 

4338 

:  4145 

3961 

3785 

7 

5447 

5201 

4967 

4746 

4535 

4334 

•  4142 

3958 

3782 

8 

5443 

5197 

4964 

4742 

4532 

4331 

!  4139 

3955 

3779 

9 

5439 

5193 

4960 

4739 

4528 

4328 

4136 

3952 

3776 

10 

5435 

5189 

4956 

4735 

4525 

4325  : 

4133 

3949 

3773 

11 

5430 

5185 

4952 

4732 

4522 

4321 

4130 

3946 

3770 

12 

5426 

5181 

4949 

4728 

4518 

4318 

4127 

3943 

3768 

13 

5422 

5177 

4945 

4724 

4515 

4315 

4124 

3940 

3765 

14 

5418 

5173 

4941 

4721 

4511 

4311 

4120 

3937 

3762 

:  15 

5414 

5169 

4937 

4717 

4508 

4308 

4117 

3934 

3759 

16 

5409 

5165 

4933 

4714 

4505 

4305 

4114  , 

3931 

3756 

17 

5405 

5161 

4930 

4710 

4501 

4302 

4111 

3928 

3753 

18 

5401 

5157 

4926 

4707 

4498 

4298  4108 

3925 

3750 

19 

5397 

5153 

4922 

4703 

4494 

4295  ’ 

4105 

3922 

3747 

20 

5393 

5149 

4918 

4699 

4491 

4292 

4102 

3919 

3745 

21 

5389 

5145 

4915 

4696 

4488 

4289 

4099  , 

.3917 

3742 

22 

5384 

5141 

4911 

4692 

4484 

4285 

4096  ; 

3914 

3739 

23 

5380 

5137 

4907 

4689 

4481 

4282 

4092  | 

3911 

3736 

24 

5376 

5133 

4903 

4685 

:  4477 

4279  1 

4089  | 

3908 

3733 

25 

5372 

5129 

4900 

4682 

4474 

4276 

4086  ( 

3905 

3730 

26 

5368 

5125 

4896 

4678 

4471 

4273  ! 

4083  1 

3902 

3727 

2  7 

5364 

5122 

4892 

4675 

4467 

4269 

4080  !  3899 

3725 

28 

5359 

5118 

i  4889 

4671 

4464 

4266  1 

4077  i 

3896 

5722 

,  29 

5355 

5114 

4885 

4668 

4460 

4263 

4074  j 

3893 

3719 

:  30 

5351 

5110 

4881 

4664 

4457 

4260  | 

4071  i 

3890  * 

3716 

TABLE  LVII. 


81 


Logistical  Logarithms . 


17' 

18' 

19' 

20' 

21' 

22' 

j  23' 

24' 

25' 

1020 

1080 

1140 

1200 

1260 

1320 

j  1380 

1440 

1500 

30" 

5351 

5110 

4881 

4664 

4457 

4260 

4071 

3890 

3716 

31 

5347 

5106 

4877 

4660 

4454 

4256 

4068 

3887 

3713 

32 

5343 

5102 

4874 

4657 

4450 

4253 

4065 

3884 

3710 

33 

5339 

5098 

4870 

4653 

4447 

4250 

4062 

3881 

3708 

34 

5335 

5094 

4866 

4650 

4444 

4247 

4059 

3878 

3705 

35 

5331 

5090 

4863 

4646 

4440 

4244 

4055 

3875 

3702 

36 

5326 

5086 

4859 

4643 

4437 

4240 

4052 

-3872 

3699 

37 

5322 

5082 

4855 

4639 

4434 

4237 

4049 

3869 

3696 

38 

5318 

5079 

4852 

4636 

4430 

4234 

4046 

3866 

3693 

39 

5314 

5075 

4848 

4632 

4427 

4231 

4043 

3863 

3691 

40 

5310 

5071 

4844 

4629 

4424 

4228 

4040 

3860 

3688 

41 

5306 

5067 

4841 

4625 

4420 

4224 

4037 

3857 

3685 

42 

5302 

5063 

4837 

4622 

4417 

4221 

4034 

3855 

3682 

43 

5298 

5059 

4833 

4618 

4414 

4218 

4031 

3852 

3679 

44 

5294 

5055 

4830 

4615 

4410 

4215 

4028 

3849 

3677 

45 

5290 

5051 

4826 

4611 

4407 

4212 

4025 

3846 

3674 

46 

5285 

5048 

4822 

4608 

4404 

4209 

4022 

3843 

3671 

47 

5281 

5044 

4819 

4604 

4400 

4205 

4019 

3840 

3668 

48 

5277 

5040 

4815 

4601 

4397 

4202 

4016 

3837 

3665 

49 

5273 

5036 

4811 

4597 

4394 

4199 

4013 

3834 

3663 

50 

5269 

5032 

4808 

4594 

4390 

4196 

4010 

3831 

3660 

51 

5265 

5028 

4804 

4590 

4387 

4193 

4007 

3828 

3657 

52 

5261 

5025 

4800 

4587 

4384 

4189 

4004 

3825 

3654 

53 

5257 

5021 

4797 

4584 

4380 

4186 

4001 

3822 

3651 

54 

5253 

5017 

4793 

4580 

4377 

4183 

3998 

3820 

3649 

55 

5249 

5013 

4789 

4577 

4374 

4180 

3995 

3817 

3646 

56 

5245 

5009 

4786 

4573 

4370 

4177 

3991 

3814 

3643 

57 

5241 

5005 

4782 

4570 

4367 

4174 

3988 

3811 

3640 

58 

5237 

5002 

4778 

4566 

4364 

4171 

3985 

3808 

3637 

59 

5233 

4998 

4775 

4563 

4361 

4167 

3982 

3805 

3635 

60 

5229  | 

4994 

4771 

4559 

4357 

4164 

3979 

3802 

3632 

*11 


82 


FABLE  LVII 


Logistical  Logarithms . 


26' 

27' 

28' 

29' 

30' 

31' 

32' 

33' 

34'  j 

1560 

1620 

1680 

1740 

1800 

1860 

1920 

1980 

2040 

0" 

3632 

3468 

3310 

3158 

3010 

2868 

2730 

2596 

2467  1 

1 

3629 

3465 

3307 

3155 

3008 

2866 

2728 

2594 

2465 

2 

3626 

3463 

3305 

3153 

3005 

2863 

2725 

2592 

2462 

3 

3623 

3460 

3302 

3150 

3003 

2861 

2723 

2590 

2460 

4 

3621 

3457 

3300 

3148 

3001 

2859 

2721 

2588 

2458 

5 

3618 

3454 

3297 

3145 

2998 

2856 

2719 

2585 

2456 

6 

3615 

3452 

3294 

3143 

2996 

2854 

2716 

2583 

2454 

7 

3612 

3449 

3292 

3140 

2993 

2852 

2714 

2581 

2452 

8 

3610 

3446 

3289 

3138 

2991 

2849 

2712 

2579 

2450 

9 

3607 

3444 

3287 

3135 

2989 

2847 

2710 

2577 

2448 

10 

3604 

3441 

3284 

3133 

2986 

2845 

2707 

2574 

2445 

11 

3601 

3438 

3282 

3130 

2984 

2842 

2705 

2572 

2443 

12 

3598 

3436 

3279 

3128 

2981 

2840 

2703 

2570 

2441 

13 

3596 

3433 

3276 

3125 

2979 

2838 

2701 

2568 

2439 

14 

3593 

3431 

3274 

3123 

2977 

2835 

2698 

2566 

2437 

15 

3590 

3428 

3271 

3120 

2974 

2833 

2696 

2564 

2435 

16 

3587 

3425 

3269 

3118 

2972 

2831 

2694 

2561 

2433 

17 

3585 

3423 

3266 

3115 

2969 

2828 

2692 

2559 

2431 

18 

3582 

3420 

3264 

3113 

2967 

2826 

2689 

2557 

2429 

19 

3579 

3417 

3261 

3110 

2965 

2824 

2687 

2555 

2426 

20 

3576 

3415 

3259 

3108 

2962 

2821 

2685 

2553 

2424 

21 

3574 

3412 

3256 

3105 

2960 

2819 

2683 

2551 

2422 

22 

3571 

3409 

3253 

3103 

2958 

2817 

2681 

2548 

2420 

23 

3568 

3407 

3251 

3101 

2955 

2815 

2678 

2546 

2418 

24 

3565 

3404 

3248 

3098 

2953 

2812 

2676 

2544 

2416 

25 

3563 

3401 

3246 

3096 

2950 

2810 

2674 

2542 

2414 

26 

3560 

3399 

3243 

3093 

2948 

2808 

2672 

2540 

2412 

27 

3557 

3396 

3241 

3091 

2946 

2805 

2669 

2538 

2410 

1  28 

3555 

3393 

3238 

3088 

2943 

2803 

2667 

2535 

2408 

1  29 

3552 

3391 

3236 

3086 

2941 

2801 

2665 

2533 

2405 

j  80 

3549 

3388 

3233 

3083 

2939 

2798 

2663 

2531 

2403 

TABLE  LVII 


83 


Logistical  Logarithms . 


26' 

27' 

28' 

29' 

30' 

31' 

32' 

33' 

34' 

1560 

1620 

1680 

1740 

1800 

1860 

1920 

1980 

2040 

30" 

3549 

3388 

3233 

3083 

2939 

2798 

2663 

2531 

2403 

31 

3546 

3386 

3231 

3081 

2936 

2796 

2660 

2529 

2401 

32 

3544 

3383 

3228 

3078 

2934 

2794 

2658 

2527 

2399 

33 

3541 

3380 

3225 

3076 

2931 

2792 

2656 

2525 

2397 

34 

3538 

3378 

3223 

3073 

2929 

2789 

2654 

2522 

2395  ( 

35 

3535 

3375 

3220 

3071 

2927 

2787 

2652. 

2520 

2393  I 

36 

3533 

3372 

3218 

3069 

2924 

2785 

2649 

2518 

2391 

37 

3530 

3370 

3215 

3066 

2922 

2782 

2647 

2516 

2389 

38 

3527 

3367 

3213 

3064 

2920 

2780 

2645 

2514 

2387 

39 

3525 

3365 

3210 

3061 

2917 

2778 

2643 

2512 

2384 

40 

3522 

3362 

3208 

3059 

2915 

2775 

2640 

2510 

2382 

41 

3519 

3359 

3205 

3056 

2912 

2773 

2638 

2507 

2380 

42 

3516 

3357 

3203 

3054 

2910 

2771 

2636 

2505 

2378 

43 

3514 

3354 

3200 

3052 

2908 

2769 

2634 

2503 

2376 

44 

3511 

3351 

3198 

3049 

2905 

2766 

2632 

2501 

2374 

45 

3508 

3349 

3195 

3047 

2903 

2764 

2629 

2499 

2372 

46 

3506 

3346 

3193 

3044 

2901 

2762 

2627 

2497 

2370 

47 

3503 

3344 

3190 

3042 

2898 

2760 

2625 

2494 

2368 

48 

3500 

3341 

3188 

3039 

2896 

2757 

2623 

2492 

2366 

49 

3497 

3338 

3185 

3037 

2894 

2755 

2621 

2490 

2364 

50 

3495 

3336 

3183 

3034 

2891 

2753 

2618 

2488 

2362 

51 

3492 

3333 

3180 

3032 

2889 

2750 

2616 

2486 

2359 

52 

3489 

3331 

3178 

3030 

2887 

2748 

2614 

2484 

2357 

53 

3487 

3328 

3175 

3027 

2884 

2746 

2612 

2482 

2355 

54 

3484 

3325 

3173 

3025 

2882 

2744 

2610 

2480 

2353 

55 

3481 

3323 

3170 

3022 

2880 

2741 

2607 

2477 

2351 

56 

3479 

3320 

3168 

3020 

2877 

2739 

2605 

2475 

2349 

57 

3476 

3318 

3165 

3018 

2875 

2737 

2603 

2473 

2347 

58 

3473 

3315 

3163 

3015 

2873 

2735 

2601 

2471 

2345 

59 

3471 

3313 

3160 

3013 

2870 

2732 

2599 

2469 

2343 

60 

3468 

3310 

3158 

3010 

2868 

2730 

2596 

2467 

2341  ! 

84 


TABLE  LVII 


Logistical  Logarithms. 


35' 

36' 

37' 

38' 

39' 

40' 

41' 

42' 

43' 

2100 

2160 

2220 

2280 

2340 

2400 

2460 

2520 

2580 

0" 

2341 

2218 

2099 

1984 

1871 

1761 

1654 

1549 

1447 

1 

2339 

2216 

2098 

1982 

1869 

1759 

1652 

1547 

1445 

2 

2337 

2214 

2096 

1980 

1867 

1757 

1650 

1546 

1443 

3 

2335 

2212 

2094 

1978 

1865 

1755 

1648 

1544 

1442 

4 

2333 

2210 

2092 

1976 

1863 

1754 

1647 

3542 

1440  1 

5 

2331 

2208 

2090 

1974 

1862 

1752 

1645 

1540 

1438  | 

6 

2328 

2206 

2088 

1972 

1860 

1750 

1643 

1539 

1437  ; 

7 

2326 

2204 

2086 

1970 

1858 

1748 

1641 

1537 

1435  ; 

8 

2324 

2202 

2084 

1968 

3856 

1746 

1640 

1535 

1433 

9 

2322 

2200 

2082 

1967 

1854 

1745 

1638 

1534 

1432 

10 

2320 

2198 

2080 

1965 

1852 

1743 

1636 

1532 

1430 

11 

2318 

2196 

2078 

1963 

1850 

1741 

1634 

1530 

1428 

12 

2316 

2194 

2076 

1961 

1849 

1739 

1633 

1528 

1427  ' 

13 

2314 

2192 

2074 

1959 

1847 

1737 

1631 

1527 

1425  j 

14 

2312 

2190 

2072 

1957  , 

1845 

1736 

1629 

1525 

1423  ! 

15 

2310 

2188 

2070 

1955 

1843 

1734 

1627 

1523 

1422  i 

.  I 

16 

2308 

2186 

2068 

1953 

1841 

1732 

1626 

1522 

1420  ' 

17 

2306 

2184 

2066 

1951 

1839 

1730 

1624 

1520 

1418 

18 

2304 

2182 

2064 

1950 

1838 

1728 

1622 

1518 

1417 

19 

2302 

2180 

2062 

1948 

1836 

1727 

1620 

1516 

1415 

20 

2300 

2178 

2061 

1946 

1834 

1725 

1619 

1515 

1413 

i 

21 

2298 

2176 

2059 

1944 

1832 

1723 

1617 

1513 

1412  j 

22 

2296 

2174 

2057 

1942 

1830 

1721 

1615 

1511 

1410  1 

23 

2294 

2172 

2055 

1940 

1828 

1719 

1613 

1510 

1408  i 

24 

2291 

2170 

2053 

1938 

1827 

1718 

1612 

1508 

1407  ; 

25 

2289 

2169 

2051 

1936 

1825 

1716 

1610 

1506 

1405  1 

26 

2287 

2167 

2049 

1934 

1823 

1714 

1608 

1504 

1403  | 

27 

2285 

2165 

2047 

1933 

1821 

1712 

1606 

1503 

1402  i 

28 

2283 

2163 

2045 

1931 

1819 

1711 

1605 

1501 

1400 

29 

2281 

2161 

2043 

1929 

1817 

1709 

1603 

1499 

1398 

30 

2279 

2159 

2041 

1927 

1816 

1707 

1601 

1498 

1397  J 

TABLE  LYII 


85 


Logistical  Logarithms. 


35' 

36' 

37' 

38' 

39' 

40' 

41 

42' 

43' 

2100 

2160 

2220 

2280 

2340 

2400 

2460 

2520 

2580 

30" 

2279 

2159 

2041 

1927 

1816 

1707 

3601 

1498 

1397 

31 

2277 

2157 

2039 

1925 

1814 

1705 

1599 

1496 

1395 

32 

2275 

2155 

2037 

1923 

1812 

1703 

1598 

1494 

1393 

33 

2273 

2153 

2035 

1921 

1810 

1702 

1596 

1493 

1392  : 

34 

2271 

2151 

2033 

1919 

1808 

1700 

1594 

1491 

1390  1 

35 

2269 

2149 

2032 

1918 

1806 

1698 

1592 

1489 

1388 

36 

2267 

2147 

2030 

1916 

1805  ! 

1696 

1591 

1487 

1387 

37 

2265 

2145 

2028 

1914 

1803  1 

i  1694 

1589 

1486 

1385 

38 

2263 

2143 

2026 

1912 

1801  ! 

!  1693 

1587 

1484 

1383  , 

39 

2261 

2141 

2024 

1910 

1799 

1691 

1585 

1482 

1382  ’ 

40 

2259 

2139 

2022 

1908 

1797 

1689 

1584. 

1481 

1380 

41 

2257 

2137 

2020 

1906 

1795 

1687 

1582 

1479 

1378 

42 

2255 

2135 

2018 

1904 

1794 

1686 

1580 

1477 

1377 

43 

2253 

2133 

2016 

1903 

1792 

1684 

1578 

1476 

1375 

44 

2251 

2131 

2014 

1901 

1790 

1682 

1577 

1474 

1373 

45 

2249 

2129 

2012 

1899 

1788 

1680 

1575 

1472 

1372 

46 

2247 

2127 

2010 

1897 

1786 

1678 

1573 

1470 

1370  1 

47 

2245 

2125 

2009 

1895 

1785 

1677 

1571 

1469 

1368 

48 

2243 

2123 

2007 

1893 

1783 

1675 

1570 

1467 

1367  | 

49 

2241 

2121 

2005 

1891 

1781 

1673 

1568 

1465 

1365 

50 

2239 

2119 

2003 

1889 

1779 

1671 

1566 

1464 

1363 

51 

2237 

2117 

2001 

1888 

1777 

1670 

1565 

1462 

1362 

52 

2235 

2115 

1999 

1886 

1775 

1668 

1563 

1460 

1360 

53 

2233 

2113 

1997 

1884 

1774 

1666 

1561 

1459 

1359 

54 

2231 

2111 

1995 

1882 

1772 

1664 

1559 

1457 

1357 

55 

2229 

2109 

1993 

1880 

1770 

1663 

1558 

1455 

1355 

'56 

2227 

2107 

1991 

1878 

1768 

1661 

1556 

1454 

1354 

57 

2225 

2105 

1989 

1876, 

1766 

1659 

1554 

1452 

1352 

58 

2223 

2103 

1987 

1875 

1765 

1657 

1552 

1450 

1350 

59 

2220 

2101 

1986 

1873 

1763 

1655 

1551 

1449 

1349 

60 

2218 

2099 

1984 

1871 

1761 

1654 

1549 

1447 

1347 

86 


TABLE  LVH 


Logistical  Logarithms. 


44' 

45' 

46' 

47' 

48' 

49' 

5  O' 

51' 

52~ 

2640 

2700 

2760 

2820 

2880 

2940 

3000 

3060 

3120 

0" 

1347 

1249 

1154 

1061 

969 

880 

792 

706 

621 

1 

1345 

1248 

1152 

1059 

968 

878 

790 

704 

620 

2 

1344 

1246 

1151 

1057 

966 

877 

789 

703 

619 

3 

1342 

1245 

1149 

1056 

965 

875 

787 

702 

617 

4 

1340 

1243 

1148 

1054 

963 

874 

786 

700 

616 

5 

1339 

1241 

1146 

1053 

962 

872 

785 

699 

615 

6 

1337 

1240 

1145 

1051 

960 

871 

783 

697 

613 

7 

1335 

1238 

1143 

1050 

959 

869 

782 

696 

612 

8 

1334 

1237 

1141 

1048 

957 

868 

780 

694 

610 

9 

1332 

1235 

1140 

1047 

956 

866 

779 

693 

609 

10 

1331 

1233 

1138 

1045 

954 

865 

777 

692 

608 

11 

1329 

1232 

1137 

1044 

953 

863 

776 

690 

606 

12 

1327 

1230 

1135 

1042 

951 

862 

774 

689 

605 

13 

1326 

1229 

1134 

1041 

950 

860 

773 

687 

603 

i  14 

1324 

1227 

1132 

1039 

948 

859 

772 

686 

602 

15 

1322 

1225 

1130 

1037 

947 

857 

770 

685 

601 

|  16 

1321 

1224 

1129 

1036 

945 

856 

769 

683 

599 

17 

1319 

1222 

1127 

1034 

944 

855 

767 

682 

598 

18 

1317 

1221 

1126 

1033 

942 

853 

766 

680 

596 

19 

1316 

1219 

1124 

1031 

941 

852 

764 

679 

595 

20 

1314 

1217 

1123 

1030 

939 

850 

763 

678 

594 

1  21 

1313 

1216 

1121 

1028 

938 

849 

762 

676 

592 

22 

1311 

1214 

1119 

1027 

936 

847 

760 

675 

591 

23 

1309 

1213 

1118 

1025 

935 

846 

759 

673 

590 

24 

1308 

1211 

1116 

1024 

933 

844 

757 

672 

588 

25 

1306 

1209 

1115 

1022 

932 

843 

756 

670 

587 

26 

1304 

1208 

1113 

1021 

930 

841 

754 

669 

585 

27 

1303 

1206 

1112 

1019 

929 

840 

753 

668 

584 

28 

1301 

1205 

1110 

1018 

927 

838 

751 

666 

583 

29 

1300 

1203 

1109 

1016 

926 

837 

750 

665 

581 

30 

1298 

1201 

1107 

1015 

924 

835 

749  1 

663 

580 

TABLE  LVIL 


87 


Logistical  Logarithms. 


44' 

'  45' 

46' 

47' 

48' 

49' 

50' 

51' 

52' 

2640 

2700 

2760 

2820 

2880 

2940 

3000 

3060 

3120 

30" 

1298 

1201 

1107 

1015 

924 

835 

749 

663 

580 

31 

1296 

1200 

1105 

1013 

923 

834 

747 

662 

579 

32 

1295 

1198 

1104 

1012 

921 

833 

746 

661 

577 

33 

1293 

1197 

1102 

1010 

920 

831 

744 

659 

576 

34 

1291 

1195 

1101 

1008 

918 

830 

743 

658 

574 

35 

1290 

1193 

1099 

1007 

917 

828 

741 

656 

573 

36 

1288 

1192 

1098 

1005 

915 

827 

740 

655 

572 

37 

1287 

1190 

1096 

1004 

914 

825 

739 

654 

570 

38 

1285 

1189 

1095 

1002 

912 

824 

737 

652 

569 

39 

1283 

1187 

1093 

1001 

911 

822 

736 

651 

568 

40 

1282 

1186 

1091 

999 

909 

821 

734 

649 

566 

41 

1280 

1184 

1090 

998 

908 

819 

733 

648 

565 

42 

1278 

1182 

1088 

996 

906 

818 

731 

647 

563 

43 

1277 

1181 

1087 

995 

905 

816 

730 

645 

562 

44 

1275 

1179 

1085 

993 

903 

815 

729 

644 

561 

45 

1274 

1178 

1084 

992 

902 

814 

727 

642 

559 

46 

1272 

1176 

1082 

990 

900 

812 

726 

641 

558 

47 

1270 

1174 

1081 

989 

899 

811 

724 

640 

557 

48 

1269  : 

1173 

1079 

987 

897 

809 

723 

638 

555 

49 

1267 

1171 

1078 

986 

896 

808 

721 

637 

554 

50 

1266 

1170 

1076 

984 

894 

806 

720 

635 

552 

51 

1264 

1168 

1074 

983 

893 

805 

719 

634 

551 

52 

1262 

1167 

1073 

981 

891 

803 

717 

633 

550 

53 

1261 

1165 

1071 

980 

890 

802 

716 

631 

548 

54 

1259 

1163 

1070 

978 

888 

801 

714 

630 

547 

55 

1257 

1162 

1068 

977 

887 

799 

713 

628 

546 

56 

1256 

1160 

1067 

975 

885 

798 

711 

627 

544 

57 

1254 

1159 

1065 

974 

884 

796 

710 

626 

543 

58 

1253 

1157 

1064 

972 

883 

795 

709 

624 

541 

59 

1251 

1156 

1062 

971 

881 

793 

707 

623 

540 

60 

1249  |  1154 

1061 

969 

880 

792 

706 

621 

539 

88 


TABLE  LVII 


Logistical  Logarithms. 


53' 

54' 

55' 

1  56< 

57' 

|  58' 

59' 

3180 

3240 

3300 

3360 

3420 

j  5480 

3540 

0 " 

539 

458 

378 

300 

223 

147 

73 

1 

537 

456 

377 

298 

221 

146 

72 

2 

536 

455 

375 

297 

220 

145 

71 

3 

535 

454 

374 

296 

219 

143 

69 

4 

533 

452 

373 

294 

218 

142 

68 

5 

532 

451 

371 

293 

216 

141 

67 

6 

531 

450 

370 

292 

215 

140 

66 

7 

629 

448 

369 

291 

214 

139 

64 

8 

528 

447 

367 

289 

213 

137 

63 

9 

526 

446 

366 

288 

211 

136 

62 

10 

525 

444 

365 

287 

210 

135 

61 

11 

524 

443 

363 

285 

209 

134 

60 

12 

522 

442 

362 

284 

208 

132 

58 

13 

521 

440 

361 

283 

206 

131 

57 

14 

520 

439 

359 

282 

205 

130 

56 

15 

518 

438 

358 

280 

204 

129 

55 

16 

517 

436 

357 

279 

202 

127 

53 

17 

516 

435 

356 

278 

201 

126 

52 

18 

514 

434 

354 

276 

200 

125 

51 

19 

513 

432 

353 

275 

199 

124 

50 

20 

512 

431 

352 

274 

197 

122 

49 

21 

510 

430 

350 

273 

196 

121 

47 

22 

509 

428 

349 

271 

195 

120 

46 

23 

507 

427 

348 

270 

194 

119 

45 

24 

506 

426 

346 

269 

192 

117 

44 

25 

505 

424 

345 

267 

191 

116 

42 

26 

503 

423 

344 

266 

190 

115 

41 

27 

502 

422 

342 

265 

189 

114 

40 

28 

501 

420 

341 

264 

187 

112 

39 

29 

499 

419 

340 

262 

186 

111 

38 

30 

498 

418 

339 

261 

185 

110  I 

36 

TABLE  LV1I 


89 


Logistical  Logarithms. 


53' 

54' 

55' 

56' 

57 

58' 

59' 

3180 

3240 

3300 

3360 

3420 

3480 

3540 

30" 

498 

418 

339 

261 

185 

110 

36 

31 

497 

416 

337 

260 

184 

109 

35 

32 

495 

415 

336 

258 

182 

107 

34 

33 

494 

414 

335 

257 

181 

106 

33 

34 

493 

412 

333 

256 

180 

105 

31 

35 

491 

411 

332 

255 

179 

104 

30 

36 

490 

410 

331 

253 

177 

103 

29 

37 

489 

408 

329 

252 

176 

101 

28 

38 

487 

407 

328 

251 

175 

100 

27 

39 

486 

406 

327 

250 

174 

99 

25 

40 

484 

404 

326 

248 

172 

98 

24 

41 

483 

403 

324 

247 

171 

96 

23 

42 

482 

402 

323 

246 

170 

95 

22 

43 

480 

400 

322 

244 

169 

94 

21 

44 

479 

399 

320 

243 

167 

93 

19 

45 

478 

398 

319 

242  • 

166 

91 

18 

46 

476 

396 

318 

241 

165 

90 

17 

47 

475 

395 

316 

239 

163 

89 

16 

48 

474 

394 

315 

238 

162 

88 

15 

49 

472 

392 

314 

237 

161 

87 

13 

50 

471 

391 

313 

235 

160 

85 

12 

51 

470 

390 

311 

234 

158 

84 

11 

52 

468 

388 

310 

233 

157 

83 

10 

53 

467 

387 

309 

232 

156 

82 

8 

54 

466 

386 

307 

230 

155 

80 

7 

55 

464 

384 

306 

229 

153 

79 

6 

56 

463 

383 

305 

228 

152 

78 

5 

57 

462 

382 

304 

227 

151 

77 

4 

58 

460 

381 

302 

225 

150 

75 

2 

59 

459 

379 

301 

224 

148 

74 

1 

60 

458 

378 

300 

223 

147 

73 

0 

*12 


90 


TABLE  LYIII 


Change  of  Moon’s  liight  Ascension  from  the  Sun . 


Time. 

38m 

39m 

40m 

41m 

42m 

43  m 

44m 

45  m 

46m 

47  m 

48m 

49  m 

50  m 

51m 

52in 

h. 

m. 

m. 

m. 

m. 

m. 

m. 

m. 

m. 

m. 

m. 

m. 

m. 

m. 

m. 

m. 

m. 

& 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

20 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

0 

40 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

0 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

1 

20 

2 

2 

2 

2 

2 

2 

2 

2 

3 

3 

3 

3 

o 

3 

3 

1 

40 

3 

3 

3 

3 

3 

3 

3 

3 

3 

3 

3 

3 

3 

4 

4 

2 

0 

3 

3 

3 

3 

3 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

2 

20 

4 

4 

4 

4 

4 

4 

4 

4 

4 

5 

5 

5 

5 

5 

5 

2 

40 

4 

4 

4 

5 

5 

5 

5 

5 

5 

5 

5 

5 

6 

6 

6 

3 

0 

5 

5 

5 

5 

5 

5 

5 

6 

6 

6 

6 

6 

6 

6 

6 

3 

20 

5 

5 

6 

6 

6 

6 

6 

6 

6 

7 

7 

7 

7 

7 

7 

3 

40 

6 

6 

6 

6 

6 

7 

7 

7 

7 

7 

7 

7 

8 

8 

8 

4 

0 

6 

6 

7 

7 

7 

7 

7 

7 

8 

8 

8 

8 

8 

8 

9 

4 

20 

7 

7 

7 

7 

8 

8 

8 

8 

8 

8 

9 

9 

9 

9 

9 

4 

40 

7 

8 

8 

8 

8 

8 

9 

9 

9 

9 

9 

10 

10 

10 

10 

5 

0 

8 

8 

8 

9 

9 

9 

9 

9 

10 

10 

10 

10 

10 

11 

11 

5 

20 

9 

9 

9 

9 

9 

10 

10 

10 

10 

10 

11 

11 

11 

11 

12 

5 

40 

9 

9 

9 

10 

10 

10 

10 

11 

11 

11 

11 

12 

12 

12 

12 

6 

0 

9 

10 

10 

10 

10 

11 

11 

11 

11 

12 

12 

12 

12 

13 

13 

6 

20 

10 

10 

11 

11 

11 

11 

12 

12 

12 

12 

13 

13 

13 

13 

14 

6 

40 

11 

11 

11 

11 

12 

12 

12 

12 

13 

13 

13 

14 

14 

14 

14 

7 

0 

11 

11 

12 

12 

12 

13 

13 

13 

13 

14 

14 

14 

15 

15 

15 

7 

20 

12 

12 

12 

13 

13 

13 

14 

14 

14 

14 

15 

15 

15 

16 

16 

7  40 

12 

12 

13 

13 

13 

14 

14 

14 

15 

15 

15 

16 

16 

16 

17 

8 

0 

13 

13 

13 

14 

14 

14 

15 

15 

15 

16 

16 

16 

17 

17 

17 

8 

20 

13 

14 

14 

14 

15 

15 

15 

.16 

16 

16 

17 

17 

17 

18 

18 

8 

40 

14 

14 

14 

15 

15 

16 

16 

16 

17 

17 

17 

18 

18 

18 

19 

9 

0 

14 

15 

.  15 

15 

16 

16 

16 

17 

17 

18 

18 

18 

19 

19 

19 

9 

20 

15 

15 

16 

16 

16 

17 

17 

18 

18 

18 

19 

19 

19 

20 

20 

9 

40 

15 

16 

16 

17 

17 

17 

18 

18 

19 

19 

19 

20 

20 

21 

21 

10 

0 

16 

16 

17 

17 

17 

18 

18 

19 

19 

20 

20 

20 

21 

21 

22 

10 

20 

16 

17 

17 

18 

18 

19 

19 

19 

20 

20 

21 

21 

22 

22 

22 

10 

40 

17 

17 

18 

18 

19 

19 

19 

20 

20 

21 

21 

22 

22 

23 

23 

11 

0 

17 

18 

18 

19 

19 

20 

20 

21 

21 

22 

22 

22 

23 

23 

24 

11 

20 

18 

18 

19 

19 

20 

20 

21 

21 

22 

22 

23 

23 

24 

24 

25 

11 

40 

18 

19 

19 

20 

20 

21 

21 

22 

22 

23 

23 

24 

24 

25 

25 

12 

0 

19 

19 

20 

20 

21 

21 

22 

22 

23 

23 

24 

24 

25 

25 

26 

TABLE  LVIII 


91 


Change  of  Moon’s  Right  Ascension  from  the  Sun. 


Time. 

53m 

54m 

55  m 

56m 

57m 

58m 

59m 

60m 

61m 

62  m 

63m 

64in 

65  m 

66  m 

h 

m. 

m. 

m. 

m. 

m. 

m. 

m. 

IT). 

m. 

m. 

m. 

m. 

m. 

m. 

m. 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

20 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

0 

40 

1 

1 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

o 

2 

1 

0 

2 

2 

2 

2 

2 

2 

2 

2 

o 

6 

3 

3 

3 

3 

3 

1 

20 

3 

3 

3 

3 

3 

3 

3 

3 

3 

3 

3 

4 

4 

4 

1 

40 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

5 

5 

2 

0 

4 

4 

5 

5 

5 

5 

5 

5 

5 

5 

5 

5 

5 

5 

2 

20 

5 

5 

5 

5 

6 

6 

6 

6 

6 

6 

6 

6 

6 

6 

2 

40 

6 

6 

6 

6 

6 

6 

7 

7 

7 

7 

7 

7 

7 

7 

3 

0 

7 

7 

7 

7 

7 

7 

7 

7 

8 

8 

8 

8 

8 

8 

3 

20 

7 

7 

8 

8 

8 

8 

8 

8 

8 

9 

9 

9 

9 

9 

3 

40 

8 

8 

8 

9 

9 

9 

9 

9 

9 

9 

10 

10 

10 

10 

4 

0 

9 

9 

9 

9 

9 

10 

10 

10 

10 

10 

10 

11 

11 

11 

4 

20 

10 

10 

10 

10 

10 

10 

11 

11 

11 

11 

11 

12 

12 

12 

4  40 

10 

10 

11 

11 

11 

11 

11 

12 

12 

12 

12 

12 

13 

13 

5 

0 

11 

11 

11 

12 

12 

12 

12 

12 

13 

13 

13 

13 

14 

14 

5 

20 

12 

12 

12 

13 

13 

13 

13 

13 

14 

14 

14 

14 

14 

15 

5 

40 

13 

13 

13 

13 

13 

14 

14 

14 

14 

15 

15 

15 

15 

15 

6 

0 

13 

13 

14 

14 

14 

14 

15 

15 

15 

15 

16 

16 

16 

16 

6 

20 

14 

14 

15 

15 

15 

15 

16 

16 

16 

16 

17 

17 

17 

17 

6 

40 

15 

15 

15 

16 

16 

16 

16 

17 

17 

17 

17 

18 

18 

18 

7 

0 

15 

16 

16 

16 

17 

17 

17 

17 

18 

18 

18 

19 

19 

19 

7 

20 

16 

16 

17 

17 

17 

18 

18 

18 

19 

19 

19 

20 

20 

20 

7 

40 

17 

17 

18 

18 

18 

19 

19 

19 

19 

20 

20 

20 

21 

21 

8 

0 

18 

18 

18 

19 

19 

19 

20 

20 

20 

21 

21 

21 

22 

22 

8 

20 

18 

19 

19 

19 

20 

20 

20 

21 

21 

22 

22 

22 

23 

23 

8 

40 

19 

19 

20 

20 

21 

21 

21 

22 

22 

22 

23 

23 

23 

24 

9 

0 

20 

20 

21 

21 

21 

22 

22 

22 

23 

23 

24 

24 

24 

25 

9 

20 

21 

21 

21 

22 

22 

23 

23 

23 

24 

24 

24 

25 

25 

26 

9 

40 

21 

22 

22 

23 

23 

23 

24 

24 

25 

25 

25 

26 

26 

27 

10 

0 

22 

22 

23 

23 

24 

24 

25 

25 

25 

26 

26 

27 

27 

27 

10  20 

23 

23 

24 

24 

25 

25 

25 

26 

26 

27 

27 

28 

28 

28 

10 

40 

24 

24 

24 

25 

25 

26 

26 

27 

27 

28 

28 

28 

29 

29 

11 

0 

24 

25 

25 

26 

26 

27 

27 

28 

28 

28 

29 

29 

30 

30 

11 

20 

25 

25 

26 

26 

27 

27 

28 

28 

29 

29 

30 

30 

31 

31 

11 

40 

26 

26 

27 

27 

28 

28 

29 

29 

30 

30 

31 

31 

32 

32 

12 

0 

26 

27 

27 

28 

28 

29 

29 

30 

30 

31 

31 

32 

32 

33 

92 


TABLE  L1X, 


Change  in  Moon’s  Declination . 


lime. 

1° 

c 

1° 

0 

10  i 

|  10/ 

20' 

31/ 

W 

50' 

Time. 

h. 

m. 

o 

/ 

O 

r 

o 

r 

' 

/ 

r 

/ 

' 

h. 

m. 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

12 

0 

0 

20 

0 

2 

0 

3 

0 

5 

0 

1 

1 

1 

1 

12 

20 

0 

40 

0 

3 

0 

7 

0 

10 

1 

1 

2 

2 

3 

12 

40 

1 

0 

0 

5 

0 

10 

0 

15 

1 

2 

2 

3 

4 

13 

0 

1 

20 

0 

7 

0 

13 

0 

20 

1 

2 

3 

4 

6 

13 

20 

1 

40 

0 

8 

0 

17 

0 

25 

1 

3 

4 

6 

7 

13 

40 

2 

0 

0 

10 

0 

20 

0 

30 

2 

3 

5 

7 

8 

14 

0 

2 

20 

0 

12 

0 

23 

0 

35 

2 

4 

6 

8 

10 

14 

20 

2 

40 

0 

13 

0 

27 

0  40 

2 

4 

7 

9 

11 

14 

40 

3 

0 

0 

15 

0 

30 

0 

45 

2 

5 

7 

10 

12 

15 

0 

3 

20 

0 

17 

0 

33 

0 

50 

3 

6 

8 

11 

14 

15 

20 

3 

40 

0 

18 

0 

37 

0 

55 

3 

6 

9 

12 

15 

15 

40 

4 

0 

0 

20 

0 

40 

1 

0 

3 

7 

10 

13 

17 

16 

0 

4 

20 

0 

22 

0 

43 

1 

5 

4 

7 

11 

14 

18 

16 

20 

4 

40 

0 

23 

0 

47 

1 

10 

4 

8 

12 

16 

19 

16 

40 

5 

0 

0 

25 

0 

50 

1 

15 

4 

8 

12 

17 

21 

17 

0 

5 

20 

0 

27 

0 

53 

1 

20 

4 

9 

13 

18 

22 

17  20 

5 

40 

0  28 

0 

57 

1 

25 

5 

9 

14 

19 

24 

17 

40 

6 

0 

0 

30 

1 

0 

1 

30 

5 

10 

15 

20 

25 

18 

0 

6 

20 

0 

32 

1 

3 

1 

35 

5 

11 

16 

21 

26 

18 

20 

6 

40 

0 

33 

1 

7 

1 

40 

6 

11 

17 

22 

28 

18 

40 

7 

0 

0 

35 

1 

10 

1 

45 

6 

12 

17 

23 

29 

19 

0 

7 

20 

0 

37 

1 

13 

1 

50 

6 

12 

18 

24 

31 

19 

20 

7  40 

0 

38 

1 

17 

1 

55 

6 

13 

19 

26 

32 

19 

40 

8 

0 

0 

40 

1 

20 

2 

0 

7 

13 

20 

27 

33 

20 

0 

8 

20 

0 

42 

1 

23 

2 

5 

7 

14 

21 

28 

35 

20 

20 

8 

40 

0 

43 

1 

27 

2 

10 

7 

14 

22 

29 

36 

20 

40 

9 

0 

0 

45 

1 

30 

2 

15 

7 

15 

22 

30 

37 

21 

0 

9 

20 

0 

47 

1 

33 

2 

20 

8 

16 

23 

31 

39 

21 

20 

9 

40 

0 

48 

1 

37 

2 

25 

8 

16 

24 

32 

40 

21 

40 

10 

0 

0 

50 

1 

40 

2 

30 

8 

17 

25 

33 

42 

22 

0 

10 

20 

0 

52 

1 

43 

2  35 

9 

17 

26 

34 

43 

22 

20 

10 

40 

0 

53 

1 

47 

2 

40 

9 

18 

27 

36 

44 

22 

40 

11 

0 

0 

55 

1 

50 

2 

45 

9 

18 

27 

37 

46 

23 

0 

11 

20 

0 

57 

1 

53 

2 

50 

9 

19 

28 

38 

47 

23 

20 

11 

40 

0 

58 

1 

57 

2 

55 

10 

19 

29 

39 

49 

23 

40 

12 

0 

1 

0 

2 

0 

3 

0 

10 

20 

30 

40 

50 

24 

0 

TABLE  LIX, 


93 


Change  of  Moon’s  Declination . 


Time. 

r| 

2' 

3' 

4' 

5' 

6' 

7' 

8' 

9' 

lime. 

h.  m. 

t 

! 

/ 

/ 

/ 

/ 

f 

h. 

in 

0  0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

12 

0 

0  20 

0 

0 

0 

0 

0 

0 

0 

0 

0 

12 

20 

0  40 

0 

0 

0 

0 

0 

0 

0 

0 

0 

12 

40 

1  0 

0 

0 

0 

0 

0 

0 

1 

1 

1 

13 

0 

1  20 

0 

0 

0 

0 

1 

1 

1 

1 

1 

13 

20 

1  40 

0 

0 

0 

1 

1 

1 

1 

1 

1 

13 

40 

i  2  0 

0 

0 

0 

1 

1 

1 

1 

1 

1 

14 

0 

2  20 

0 

0 

1 

1 

1 

1 

1 

2 

2 

14 

20 

2  40 

0 

0 

1 

1 

1 

1 

2 

2 

2 

14 

40 

3  0 

0 

0 

1 

1 

1 

1 

2 

2 

2 

15 

0 

3  20 

0 

1 

1 

1 

1 

2 

2 

2 

2 

15 

20 

3  40 

0 

1 

1 

1 

2 

2 

2 

2 

3 

15 

40 

4  0 

0 

1 

1 

1 

2 

2 

2 

3 

3 

16 

0 

4  20 

0 

1 

1 

1 

2 

2 

3 

3 

3 

16 

20 

4  40 

0 

1 

1 

2 

2 

2 

3 

3 

3 

16 

40 

5  0 

0 

1 

1 

2 

2 

2 

3 

3 

4 

17 

0 

5  20 

0 

1 

1 

2 

2 

3 

3 

4 

4 

17 

20 

5  40 

0 

1 

1 

2 

2 

3 

3 

4 

4 

17 

40 

6  0 

1 

1 

1 

2 

2 

3 

3 

4 

4 

18 

0 

6  20 

1 

1 

2 

2 

3 

3 

4 

4 

5 

18 

20 

6  40 

1 

1 

2 

2 

3 

3 

4 

4 

5 

18 

40 

7  0 

1 

1 

2 

2 

3 

3 

4 

5 

5 

19 

0 

7  20 

1 

1 

2 

2 

3 

4 

4 

5 

5 

19 

20 

7  40 

1 

1 

2 

3 

3 

4 

4 

5 

6 

19  40 

8  0 

1 

1 

2 

3 

3 

4 

5 

5 

6 

20 

0 

8  20 

1 

1 

2 

3 

3 

4 

5 

6 

6 

20 

20 

8  40 

1 

1 

2 

3 

4 

4 

5 

6 

6 

20 

40 

9  0 

1 

1 

2 

3 

4 

4 

5 

6 

7 

21 

0 

9  20 

1 

2 

2 

3 

4 

5 

5 

6 

7 

21 

20 

9  40 

1 

2 

2 

3 

4 

5 

6 

6 

7 

21 

40 

10  0 

1 

2 

2 

3 

4 

5 

6 

7 

7 

22 

0 

10  20 

1 

2 

3 

3 

4 

5 

6 

7 

8 

22 

20 

10  40 

1 

2 

3 

4 

4 

5 

6 

7 

8 

22 

40 

11  0 

1 

2 

3 

4 

5 

5 

6 

7 

8 

23 

0 

11  20 

1 

2 

3 

4 

5 

6 

7 

8 

8 

23 

20 

11  40 

1 

2 

3 

4 

5 

6 

7 

8 

9 

23 

40 

12  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

24  00 

94 


TABLE  LX 


For  the  Aberration  of  a  Star  in  Right  Ascension  and 
Declination . 


Argument.  Sun’s  true  Longitude. 


0 

VIs 

1* 

Vll* 

II* 

VIII* 

Log.  a 

X 

Log.  a 

X 

Log.  a 

X 

— 

+ 

— 

4* 

— 

+ 

0° 

1.2690 

0°  0' 

1.2790 

2°1T 

1.2977 

2°  6' 

30' 

1 

1.2690 

0  5 

1.2796 

2  14 

1.2983 

2  3 

29 

2 

1.2691 

0  11 

1.2802 

2  16 

1.2988 

2  0 

28 

3 

1.2692 

0  16 

1.2808 

2  18 

1.2993 

1  57 

27 

4 

1.2692 

0  22 

1.2815 

2  20 

1.2998 

1  54 

26 

5 

1.2693 

0  27 

1.2821 

!  2  21 

1.3003 

1  51 

25 

6 

1.2695 

0  32 

1.2827 

2  23 

1.3008 

1  47 

24 

7 

1-2696 

0  37 

1.2834 

2  24 

1.3012 

1  44 

23 

8 

1.2698 

0  43 

1.2840 

2  25 

1.3017 

1  40 

22 

9 

1.2700 

0  48 

1.2847 

2  26 

1.3021 

1  36 

21 

10 

1.2703 

0  53 

1.2853 

2  27 

1.3025 

1  32 

20 

11 

1.2705 

0  58 

1.2860 

2  28 

1.3028 

1  28 

19 

12 

1.2708 

1  3 

1.1866 

2  28 

1.3032 

1  24 

18 

13 

1.2711 

1  8 

1.2873 

2  28 

1.3036 

1  20 

17 

14 

1.2714 

1  12 

1.2879 

2  28 

1.3039 

l  16 

16 

15 

1.2718 

1  17 

1.2886 

2  28 

1.3042 

1  11 

15 

16 

1.2721 

1  22 

1.2892 

2  28 

1.3045 

1  7 

14 

17 

1.2725 

1  26 

1.2899 

2  27 

1.3048 

1  3 

13 

i  18 

1.2729 

1  30 

1.2905 

2  27 

1.3050 

0  58 

12 

i  19 

1.2733 

1  34 

1.2912 

2  26 

1.3053 

0  53 

11 

20 

1.2738 

1  39 

1.2918 

2  25 

1.3055 

0  49 

10 

21 

1.2742 

1  42 

1.2924 

2  24 

1.3057 

0  44 

9 

|  22 

1.2747 

1  46 

1.2931 

2  22 

1.3059 

0  39 

8 

23 

1.2752 

1  50 

1.2938 

2  21 

1.3060 

0  34 

7 

24 

1.2757 

1  53 

1.2944 

2  19 

1.3061 

0  30 

6 

25 

1.2762 

1  57 

1.2949 

2  17 

1.3063 

0  25 

5 

26 

1.2768 

2  0 

1.2956 

2  15 

1.3064 

0  20 

4 

-  27 

1.2773 

2  3 

1.2961 

2  13 

1.3064 

0  15 

3 

28 

1.2779 

2  6 

1.2966 

2  11 

1,3065 

0  10 

2 

29 

1.2785 

2  9 

1.2972 

2  8 

1 .3065 

0  5 

1 

30 

1.2790 

2  11 

1.2977 

2  6 

1.3065 

0  0 

0 

Log1,  a 

X 

Log.  a 

X 

Log.  a 

X 

Vs 

XIs 

IV* 

X* 

III* 

IXs 

TABLE  LXI, 


95 


For  the  Aberration  of  a  Star  in  Right  Ascension  and 
Declination . 

Argument.  Sun’s  Longitude,  more  or  less  the  Star’s  Declination. 


0* 

Is 

11* 

IIP 

IVs 

Vs 

— 

— 

— 

-f 

+ 

+ 

0° 

4".03 

3".49 

2".02 

0".00 

2 ''.02 

3".49 

30° 

1 

4.03 

3.46 

1.95 

0.07 

2.08 

3.53 

29 

2 

4.03 

3.42 

1.89 

0.14 

2.14 

3.56 

28 

3 

4.03 

3.38 

1.83 

0.21 

2.20 

3.59 

27 

4 

4.02 

3.34 

1.77 

0.28 

2.26 

3.63 

26 

5 

4.02 

3.30 

1.70 

0.35 

2.31 

3.66 

25 

6 

4.01 

3.26 

1.64 

0.42 

2.37 

3.68 

24 

7 

4.00 

3.22 

1.58 

0.49 

2.43 

3.71 

23 

8 

3.99 

3.18 

1.51 

0.56 

2.48 

3.74 

22 

9 

3.98 

3.13 

1.45 

0.63 

2.54 

3.77 

21 

10 

3.97 

3.09 

1.38 

0.70 

2.59 

3.79 

20 

11 

3.96 

3.04 

1.31 

0.77 

2.65 

3.81 

19 

12 

3.95 

3.00 

1.25 

0.84 

2.70 

3.84 

18 

13 

3.93 

2.95 

1.18 

0.91 

2.75 

3.86 

17 

14 

3.91 

2.90 

1.11 

0.98 

2.80 

3.88 

16 

15 

3*90 

2.85 

1.04 

1.04 

2.85 

3.90 

15 

16 

3.88 

2.80 

0.98 

1.11 

2.90 

3.91 

14 

17 

3.86 

2.75 

0.91 

1.18 

2.95 

3.93 

13 

18 

3.84 

2.70 

0.84 

1.25 

3.00 

3.95 

12 

19 

3.81 

2.65 

0.77 

1.31 

3.04 

3.96 

11 

20 

3.79 

2.59 

0.70 

1.38 

3.09 

3.97 

10 

21 

3.77 

2.54 

0.63 

1.45 

3.13 

3.98 

9 

22 

3.74 

2.48 

0.56 

1.51 

3.18 

3.99 

8 

23 

3.71 

2.43 

0.49 

1.58 

3.22 

4.00 

7 

24 

3.68 

2.37 

0.42 

1.64 

3.26 

4.01 

6 

25 

3.66 

2.31 

0.35 

1.70 

3.30 

4.02 

5 

26 

3.63 

2.26 

0.28 

1.77 

3.34 

4.02 

4 

27 

3.59 

2.20 

0.21 

1.83 

3.38  : 

4.03 

3 

28 

3.56 

2.14 

0.14 

1.89 

3.42 

4.03 

2 

29 

3.53 

2.08 

0.07 

1.95 

3.46 

4.03 

1 

30 

3.49 

2.02 

0.00 

2.02 

3.49 

4.03 

0 

— 

— 

— 

4~ 

4- 

■+■ 

XIs 

X* 

IX* 

VIIIs 

VIIs 

VP 

96 


TABLE  LX1I 


For  the  «5V* station  in  Right  Jlscension  and  Declination . 

Argument.  Mean  Longitude  of  Moon’s  Ascending  Node. 


0* 

VI* 

%  Is 

VII* 

II* 

Vlll* 

Log.  b 

B 

j  Log- 

B 

Log.  b 

B 

0° 

0.9844 

0°  0' 

0.9588 

6°  45' 

0.8960 

7°  48' 

30° 

1 

0.9844 

0  15 

0.9571 

6  54 

0.8939 

7  40 

29 

2 

0.9843 

0  31 

0.9554 

7  3 

0.8917 

7  32 

28 

3 

0.9842 

0  46 

0.9536 

7  12 

0.8896 

7  23 

27 

4 

0.9840 

1  1 

0.9518 

7  20 

0.8875 

7  14 

26 

5 

0.9837 

1  16 

0.9500 

7  28 

0.8854 

7  4 

25 

6 

0.9834 

1  32 

0.9481 

7  36 

0.8834 

6  53 

24 

7 

0.9830 

1  47 

0.9462 

7  43 

0.8814 

6  42 

23 

8 

0.9825 

2  2 

0.9442 

7  49 

0.8795 

6  29 

22 

9 

0.9821 

2  17 

0.9422 

7  55 

0.8776 

6  17 

21 

10 

0.9815 

2  31 

0.9402 

8  1 

0.8758 

6  3 

20 

11 

0.9809 

2  46 

0.9382 

8  6 

0.8740 

5  49 

19 

12 

0.9802 

3  1 

0.9361 

8  10 

0.8723 

5  35 

18 

13 

0.9795 

3  15 

0.9340 

8  14 

0.8707 

5  20 

17 

14 

0.9787 

3  29 

0.9318 

8  17 

0.8692 

5  4 

16 

15 

0.9779 

3  45 

0.9297 

8  20 

0.8677 

4  48 

15 

16 

0.9770 

3  57 

0.9275 

8  23 

0.8663 

4  31 

14 

17 

0.9760 

4  11 

0.9253 

8  24 

0.8650 

4  14 

13 

I  18 

0.9750 

4  24 

0.9231 

8  25 

0.8637 

3  56 

12 

19 

0.9739 

4  37 

0.9208 

8  25 

0.8625 

3  38 

11 

20 

0.9728 

4  50 

0.9186 

8  25 

0.8615 

3  20 

10 

21 

0.9716 

5  3 

0.9163 

8  24 

0.8605 

3  1 

9 

22 

0.9704 

5  16 

0.9140 

8  23 

0.8596 

2  41 

8 

23 

0.9691 

5  28 

0.9118 

8  21 

0.8588 

2  22 

7 

24 

0.9678 

5  40 

0.9095 

8  18 

0.8582 

2  2 

6 

25 

0.9664 

5  51 

0.9072 

8  15 

0.8576 

1  42 

5 

26 

0.9650 

6  3 

0.9050 

8  11 

0.8571 

1  22 

4 

27 

0.9635 

6  14 

0.9027 

8  6 

0.8568 

1  2 

3 

28 

0.9620 

6  24 

0.9005 

8  1 

0.8565 

0  41 

2 

29 

0.9604 

6  35 

0.8983 

7  55 

0.8563 

0  21 

1 

;  30 

0.9588 

6  45 

0.8960 

7  48 

0.8563 

0  0 

0 

_ 

4* 

_ 

+ 

_ 

4_ 

Log.  b 

B 

Log.  b 

B 

Log.  b 

B 

V* 

XIs 

IV* 

X* 

III* 

IX* 

TABLE  LXIII.  97 

For  the  Nutation  in  Right  Ascension  and  Declination . 

Argument.  Mean  Longitude  of  Moon’s  Ascending  Node. 


0* 

1* 

iis 

Hi* 

IV* 

V* 

0° 

0".00 

8".2  7 

14". 33 

16"54 

14"33 

8  ".27 

30° 

1 

0.29 

8.52 

14.47 

16.54 

14.18 

8.02 

29 

2 

0.'58 

8.77 

14.61 

16.53 

14.03 

7.77 

28 

3 

0.87 

9.01 

14.74 

16.52 

13.83 

7.51 

27 

4 

1.15 

9.25 

14.87 

16.50 

13.72 

7.25 

26 

5 

1.44 

9.49 

14.99 

16.48 

13.55 

6.99 

25 

6 

1.75 

9.72 

15.11 

16.45 

13.38 

6.73 

24 

7 

2.02 

9.96 

15.23 

16.42 

13.21 

6.46 

23 

8 

2.30 

10.19 

15.34 

16.38 

13.04 

6.20 

22 

9 

2.59 

10.41 

15.45 

16.34 

12.86 

5.93 

21 

10 

2.87 

10.63 

15.55 

16.29 

12.67 

5.66 

20 

11 

3.16 

10.85 

15.64 

16.24 

12.49 

5.39 

19 

12 

3.44 

11.07 

15.73 

16.18 

12.30 

5.11 

18 

13 

3.72 

11.28 

15.82 

16.12 

12.10 

4.84 

17 

14 

4.00 

11.49 

15.90 

16.05 

11.90 

4.56 

16 

15 

4.28 

11.70 

15.98 

15.98 

11.70 

4.28 

15 

16 

4.56 

11.90 

16.05 

15.90 

11.49 

4.00 

14 

17 

4.84 

12.10 

16.12 

15.82 

11.28 

3.72 

13 

18 

5.11 

12.30 

16.18 

15.73 

11.07 

3.44 

12 

19 

5.39 

12.49 

16.24 

15.64 

10.85 

3.16 

11 

20 

5.66 

12.67 

16.29 

15.55 

10.63 

2.87 

10 

21 

5.93 

12.86 

16.34 

15.45 

10.41 

2.59 

9 

22 

6.20 

13.04 

16.38 

15.34 

10.19 

2.30 

8 

23 

6.46 

13.21 

16.42 

15.23 

9.96 

2.02 

7 

24 

6.73 

13.38 

16.45 

15.11 

9.72 

1.75 

6 

25 

6.99 

13.55 

16.48 

14.99 

9.49 

1.44 

5 

26 

7.25 

13.72 

16.50 

14.87 

9.25 

1.15 

4 

27 

7.51 

13.83 

16.52 

14.74 

9.01 

0.87 

3 

28 

7.77 

14.03 

16.53 

14.61 

8.77 

0.58 

2 

29 

8.02 

14.18 

16.54 

14.47 

8.52 

0.29 

1 

30 

8.27 

14.33 

16.54 

14.33 

8.27 

0.00 

0 

4- 

!  4- 

+ 

+ 

4“ 

4- 

XI* 

X* 

IX* 

VIII* 

VII* 

VI* 

13* 


98 


TABLE  LXIV, 


Semidiurnal  Arcs  for  39°  57'  North  Latitude. 


North  Declination. 

Sooth  Declination. 

O' 

20' 

40' 

0' 

2  O' 

40' 

h.  m. 

h.  m. 

h.  m. 

h. 

m. 

h. 

m. 

h. 

m. 

0° 

6  0 

6  1 

6  2 

6 

0 

5 

59 

5 

58 

1 

6  3 

6  4 

6  6 

5 

57 

5 

56 

5 

54 

2 

6  7 

6  8 

6  9 

5 

53 

5 

52 

5 

51 

cy 

O 

6  10 

6  11 

6  12 

5 

50 

5 

49 

5 

48 

4 

6  13 

6  15 

6  16 

5 

47 

5 

45 

5 

44 

5 

6  17 

6  18 

6  19 

5 

43 

5 

42 

5 

41 

6 

6  20 

6  21 

6  22 

5 

40 

5 

39 

5 

38 

7 

6  24 

6  25 

6  26 

5 

36 

5 

35 

5 

34 

8 

6  27 

6  28 

6  29 

5 

33 

5 

32 

5 

31 

9 

6  30 

6  32 

6  33 

5 

30 

5 

28 

5 

27 

10 

6  34 

6  35 

6  36 

5 

26 

5 

25 

5 

24 

11 

6  37 

6  39 

6  40 

5 

23 

5 

21 

5 

20 

12 

6  41 

6  42 

6  43 

5 

19 

5 

18 

5 

17 

13 

6  45 

6  46 

6  47 

5 

15 

5 

14 

5 

13 

14 

6  48 

6  49 

6  51 

5 

12 

5 

11 

5 

9 

15 

6  52 

6  53 

6  54 

5 

8 

5 

7 

5 

6 

16 

6  56 

6  57 

6  58 

5 

4 

5 

3 

5 

2 

17 

6  59 

7  1 

7  2 

5 

1 

4 

59 

4 

58 

18 

7  3 

7  4 

7  6 

4 

57 

4 

56 

4 

54 

19 

7  7 

7  8 

7  10 

4  53 

4 

52 

4 

50 

20 

7  11 

7  12 

7  14 

4 

49. 

4 

48 

4 

46 

21 

7  15 

7  16 

7  18 

4 

45 

4 

44 

4 

42 

22 

7  19 

7  21 

7  22 

4 

41 

4 

39 

4  38 

23 

7  23 

7  25 

7  26 

4  37 

4 

35 

4 

34 

24 

7  28 

7  29 

7  30 

4  32 

4 

31 

4 

30 

25 

7  32 

7  33 

7  35 

4 

28 

4 

27 

4 

25 

26 

7  36 

7  38 

7  40 

4 

24 

4 

22 

4 

20 

27 

7  41 

7  43 

7  44 

4 

19 

4 

17 

4 

16 

28 

7  46 

7  47 

7  49 

4 

14 

4 

13 

4 

11 

29 

7  51 

7  52 

7  54 

4 

9 

4 

8 

4 

6 

30 

7  56 

7  57 

7  59 

4 

4 

4 

3 

4 

1 

THE  END 


ERRATA. 


Page  Line 


14 

18 

u 

19 

15 

11 

23 

19 

24 

13 

26 

25 

27 

4 

29 

20 

34 

8 

35 

20 

36 

19 

61 

17 

75 

17 

u 

22 

76 

15 

96 

19 

97 

3 

98 

5 

104 

13 

<c 

14 

110 

11 

112 

7 

114 

13 

115 

5 

119 

16 

123 

7 

124 

10 

127 

6 

For 


tan  n' 


read 


tan  n’f 


r  r 

For  PM  =  PZ  f  ZM  =  PZ  +  »"  +  r",  read  PM 
=  ZM  —  PZ  r  n"  +  r"  —  PZ. 

For  Formule,  read  Formulae. 

For  20°,  read  90°. 

r  T  '3T  j  1 1 

For  — ,  read  — . 

R  ST 

For  PEp2,  read  PEpQ. 

R' 

R 


F  R  ,  R 

±or  — ,  read 


For  ecliptic,  read  equator. 

For  PSL,  read  P'SL. 

For  of  the  star,  read  of  the  sun  or  star. 

For  L  tan  read  cos  L  tan  a. 

For  ang.  AEL.  read  ang.  ACL. 

For  RL,  read  BL. 

For  ,  read  -f . 

For  .83*7,  read  .08 3^. 

For  sine  of  the  elongation,  read  sine  of  half  the 
elongation. 

For  EE,  read  EC. 

For  times,  read  time. 

For  1  c°s  MCO  read  1  cos  MCO 
sin  MCO  cos  MCO 

For  sin  MEO,  read  sin  MCO. 

n  sin  5  E  ,  sin  E 

For  - read  . - 

sin  I  F  sin  F 
For  latitude,  read  polar  distance. 

For  sin  ( — n  4-  n),  read  sin  (L  —  n-f  n). 

For  sin  (A  4.  n  —  y),  read  sin  (A  4-  ?r  —  y). 

For  these  will  be  an  ecliptic,  read  there  wiil  be  an 
eclipse. 

For  3684.3,  read  3683.3. 

For  she,  read  the. 

For  is  latitude,  read  is  the  latitude. 


ERRATA . 


Page  Line 

130  25  nad*=Xlll  +  d  —  a  cos  1) 

2d 

“  26  read  =  ^  ^  cosil'.._. 

d 

•133  23  For  CES,  read  CEQ 

142  34  For  h  (o~c.tat?il,  read  2da  ~  CA™  x. 

1  +  tan2 1  1  +  tan2 1 

149  7  For  PP,  read  PP'. 

160  3  For  S v,read  S p. 

162  13  For  EQS,  read  QES. 

164  9  For  geocentrical,  read  geocentric. 

175  24  For  52  and  54,  read  54  and  56. 

192  19  For  porabol a,  read  parabola. 

197  9  For  mutation,  read  nutation. 

204  34  For  semi  diameter,  read  semidiameters. 

234  6  For  Schehallier,  read  Shehallien. 

299  18  For  logarithm  of  B,  read  logarithm  B. 

302  8  For  Take  the  difference  between  the  moon’s  longi¬ 

tude  and  the  longitude  of  the  nonagesimal  degree, 
and  call  it  D,  read  Subtract  the  longitude  of  the 
nonagesimal  from  the  moon’s  longitude,  increasing 
the  latter  by  360°,  if  necessary,  and  when  the  re¬ 
mainder  is  less  than  180°,  it  is  the  moon’s  distance 
to  the  east  of  the  nonagesimal,  which  call  D;  but 
when  the  remainder  is  greater  than  180°,  subtract 
it  from  360°,  and  the  second  remainder  will  be  D, 
the  moon’s  distance  to  the  west  of  the  nonagesimal. 

303  4  For  Add  p  to,  &c.  read  Add  p  to  the  moon' s  true 

longitude,  when  the  moon  is  to  the  east  of  the 
nonagesimal;  but  subtract ,  when  it  is  to  the  acesf, 
and  the  result  will  be  the  apparent  longitude. 

304  14  For  The  parallax  in  longitude,  p ,  & c.  substitute  the 

same  as  directed  the  last  above. 

814  11  For  time  of  new  moon,  read  time  of  new  moon,  reck¬ 

oned  from  the  noon  or  midnight  to  which  the 
second  distance  corresponds. 


Plate 


Plate  2. 


Yciauf  £])e?l*Jcer  S<£ 


Pla/e  2. 


Plate  3 . 


•/WWw  JV. 


8 . 

— —  r 

r  a 

d 

Fuf-27. 

\ 

M 

B  Ff 

if 

F 

5  C' 

P  S' 

Flute  /. 


/v 


Plot,- 


Plate  (>'. 


r/.it. 


Fig 


